(*) Introduction
Observe the example given below :
1) Euclid, Pythagoras, Gauss, Leibnitz, Aryabhatta, Bhaskar.
2) a,e,i,o,u
3) Happy, sad, angry, anxious, joyful, confused.
4) Cricket, football, kho-kho, kabaddi, basketball
5) 1,3,5,7,9......
What do you observe?
Example 1 is a collection of name of some mathematicians.
Example 2 is a collection of vowel letters in the English alphabet
Example 3 is a collection of feelings.
We see that the name/items/objects in each example have something in common.
i.e., they form a collection .
We come across collections in mathematics too.
For example:
> natural numbers,
>prime numbers,
>quadrilateral in a plane etc.
All examples seen so far are well defined collection of objects or ideas.
A well defined collection of objects or ideas is known as a Set.
(*) Well defined Sets
What do we mean when we say that a set is a well defined collection of objects. Well defined means that :
1) All the objects in the set should have a common feature or property; and
2) It should be possible to decide whether any given object belongs to the set or not.
Let us understand ' well defined' through some examples.
Consider the statement : The collection of all tall students in your class.
What difficulty is caused by this statement? Here, who is tall is not clear.
*) Richa decides that all students taller than her are tall. Her set has five students.
*) Yashodhara also decides that tall means all students taller than her. her set has ten students.
*) Ganapati decides that tall means every student whose height is more than 5 feet. His set has 3 students.
We find that different people get different collections. So, this collection is not well defined.
Now consider the following statement : The collection of all students in your class who are taller than 5 feet 6 inches.
In this case, Richa, Yashodhara and Ganapati, all will get the same collection . So, the collection forms a well defined set.
(*) Naming of Sets and Elements of a Set
We usually denote a set by upper case letters, A,B,C,X,Y,Z etc. A few examples of sets in mathematics are given below.
*) The set of all Natural numbers is denoted by N.
*) The set of all Integers is denoted by Z.
*) The set of all Rational numbers is denoted by Q.
*) The set of all Real numbers is denoted by R.
Notice that all the sets given above are well defined collections because given a number we can decide whether it belongs to the set or not.
Let us see some more examples of elements.
Suppose we define a set as all days in a week, whose name begins with T.
Then we know that Tuesday and Thursday are part of the set but Monday is not.
We say that Tuesday and Thursday are elements of the set of all days in a week starting with T.
Consider some more examples :
(!) We know that N usually stands for the set of all natural numbers. Then 1,2,3.....are elements of the set. But 0 is not an element of N.
(!!) Let us consider the set B, of quadrilaterals.
B={ square, rectangle, rhombus, parallelogram}
Can we put triangle, trapezium or cone in the above set 'B'?
No, a triangle and cone are cannot be members of B.
But a trapezium can be a member of the set B.
So, we can say that an object belonging to a set is knows as a member/ element of the set. We
use the symbol ∈ to denote 'belongs to'.
So 1∈ N means that 1 belongs to N. Similarly 0 ∌ N means that 0 does not belong to N.
There are various ways in which we can write sets.
For Example, we have the set of all vowel letters in the English alphabet. Then, we can write :
(!) V = {a,e,i,o,u}, Here,we list down all the elements of the set between chain/curly brackets. This is called the roster form of writing sets.
In roster form, all elements of the set are written, separated by commas, within curly brackets.
(!!) V= { x:x is a vowel letter in the English alphabet}
or V = {x/x is a vowel letter in the English alphabet}
This way of writing a set is known as the set builder form.
Here , we use symbol x(or any other symbol y,z etc.,) for the element of the set. This is followed by a colon (or a vertical line) , after which we write the characteristic property possessed by the elements of the set.
The whole is enclosed within curly brackets.
Let C = { 2,3,5,7,11}, a set of prime numbers less than 13. this set can be denoted as :
C = {x/x is a prime number less than 13} or
C = {x:x is a prime number less than 13}.
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Example-1. Write the following in roster and set builder form.
(!) The set of all natural numbers which divide 42.
sol) Let B be the set of all natural numbers which divide 42. Then, we can write :
B = { 1,2,3,6,7,14,21,42}= Roster form
B ={x:x is a natural number which divides 42} = Set builder form
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(!!) The set of natural numbers which are less than 10.
Sol) let A be the set of all natural numbers which are less than 10. Then, we can write :
A={1,2,3,4,5,6,7,8,9}= Roster form
B={x:x is a natural number which is less than 10} = set builder form
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Note :
(!) In roster form, the order in which the elements are listed is immaterial. Thus, in example 1, we can also write {1,3,7,21,2,6,4,42}
(!!) While writing the elements of a set in roster form, an element is not repeated. For example, the set of letters forming the word "SCHOOL" is {S,C,H,O,,L} and not {S,C,H,O,O,L}
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Example-2, Write the set B={x:x is a natural number and x^2<40 } in the roster form.
sol) We look at natural numbers and their squares starting from 1. When we reach 7, the square is 49 which is greater than 40. The required numbers are 1,2,3,4,5,6.
So, the given set in the roster form is
B = { 1,2,3,4,5,6}.
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Exercise-2.1
1) Which of the following are sets? Justify your answer?
(1) The collection of all the months of a year beginning with the letter "J".
Sol) This is a set .why?
Because the months starts with a letter "j" are ( January,June,July) ..
We also know no extra month can be added or removed or replaced from year calendar .
Thus , it is well defined set.
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(2) The collection of ten most talented writers of India.
sol) This is not a set. why?
Because the top ten writers names today can be replace by other writers name in future.
Thus , it is not well-defined set.
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(3) A team of 11 best cricket batsmen of the world.
Sol) The names of 11 best batsmen can replace by other batsman in future.
Thus it is not a set.
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(4) The collection of all boys in your class.
Sol) This is a set
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(5) The collection of all even integers.
Sol) This is a set.
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2) If A = {0,2,4,6 }, B= {3,5,7} and C= {p,q,r} then fill the appropriate symbols,
1) 0.....∈......A (Since 0 is an element of A)
2) 3....∉.......C (Since 3 is not an element of C)
3) 4.....∉......B (Since 4 is not an element in B)
4) 8.....∉......A (Since 8 is not an element in A)
5) p....∈.......C (Since p is an element in C)
6) 7....∈.......B (Since 7 is an element in B)
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3) Express the following statements using symbols.
(1) The elements X does not belong to "A"
sol) x ∉ A
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(2) "d" is an element of the set "B"
sol) d ∈ B
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(3) '1" belongs to the set of Natural numbers N.
sol) 1 ∈ N
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(4) "8" does not belong to the set of prime numbers "P"
sol) 8 ∉ P
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4) State whether the following statements are true of false.
(1) 5 does not belongs to { prime numbers }
Sol) False . As we know "5' can only be divided by "1" and "itself" .So it is a prime numbers.
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(2) S = {5,6,7 } implies 8 belongs to S
Sol) False.
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(3) -5 does not belongs to "W" where "W" is the set of whole numbers.
Sol) True. As we know whole number does not contain fractional, or decimal or negative numbers..It contain only positive integers.(0,1,2,3,...)
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(4) 8/11 belongs to Z where "Z" is the set of integers.
Sol) False . An integer does not contain fractional part.
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5. Write the following sets in roster form.
(1) B = { x : x is a natural number less than 6 }
Sol) B = {1,2,3,4,5}
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(2) C = { x : x is a two-digit natural number such that the sum of its digits is 8}
Sol) C = {17,26,35,44,53,62,71}
((17) 1+7=8,..(26)2+6=8,...so on)
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(3) D = {x : x is a prime number which is a divisor of 60 }
Sol) D = {2,,3,5 }
(60/2=30......60/5 =12...60/3=20...)
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(4) E = {the set of all letters in the word BETTER }
Sol) E = {B,E,T,R}
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6) Write the following sets in the set-builder form.
(1) {3,6,9,12 }
Sol) if we observe the set we can clearly see it is in multiple of "3"
= ( 3*1=3.....3 *2= 6,...3*3=9.....3*4=12)
={3,6,9,12}
Set-builder:-
A = { x :x is a multiple of 3 and less than 13 }
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(2) { 2,4,8,16,32 }
Sol ) it is in power of 2^x form
= (2^1=2....2^2=4...2^3=8.....2^4=16....2^5=32)=
{2,4,8,16,32}
Set-builder :
A = { x : x is in power of 2^x and x is less than 6}
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(3) { 5,25,125,625}
Sol) It is in 5^x form
= ( 5^1 = 5.....5^2=25.....5^3= 125...5^4=625}=
{5,25,125,625}
Set-builder :-
A= { x :x is in power of 5^x and x is less than 5}
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(4) {1,4,9,16,25,.....100}
Sol) (1^2=1....2^2=4....3^2=9....4^2=16....5^2=25.....10^2=100)
Set-builder:
A = { x :x is a square of natural number but not greater than 10}
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7) List all the elements of the following sets in roster form.
(1) A= { x: x is a natural number greater than 50 but less than 100}
Sol) A = { 51,52,53,54,55..............97,98,99}
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(2) B = { x: x is an integer, x^2 =4 }
Sol) B= {-2 , 2}
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(3) D = { x: x is a letter in the word "LOYAL" }
sol) D = { L,O,Y,A}
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8) Match the roster form with set-builder form.
(1) {1,2,3,6}
(a) {x:x is prime number and a divisor of 6}
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(2) {2,3}
(b) {x:x is an odd natural number smallest than 10}
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(3) {M,A,T,H,E,I,C,S}
(c){x:x is a natural number and divisor of 6}
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(4) {1,3,5,7,9}
(d) {x:x is a letter of the word MATHEMATICS }
SOLUTION:-
1. { 1,2,3,6 } => { x: x is a natural number and divisor of 6}
2. {2,3} => { x:x is prime number and a divisor of 6}
3. {M,A,T,H,E,I,C,S} => { x:x is a letter of the word MATHEMATICS}
4. {1,3,5,7,9} => {x:x is an odd natural number less than 10}
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(*) Types of Set
Let us consider the following examples of sets :
(!) A = { x:x is natural number smaller than 1}
(!!) D = {x:x is a odd prime number divisible by 2}
How many elements are there in A and D? we find that there is no natural number which is smaller than 1. So set A contains no elements or we say that A is an empty set.
Similarly, there are no prime numbers that are divisible by 2 . So, D is also an empty set.
A set which does not contain any element is called an empty set, or a Null set, or a void set. Empty set is denoted by the symbol ∅or { }
Here are some more examples of empty sets.
(!) A = { x:1<x<2, x is a natural number}
(!!) B = {x:x^2=0 and x is a rational number}
(!!!) D = {x:x^2 = 4, x is odd }
Note : ∅ and {0} are two different sets. {0} is a set containing the single element 0 while { } is null set.
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Finite & Infinite sets
Now consider the following sets :
(!) A = { the students of your school}
(!!) L = {p,q,r,s}
(!!!) B = {x :x is an even number}
(!v) J = { x : x is a multiple of 7 }
In (!), the number of elements will be the number of students in your school.
In (!!) the number of elements in set L is 4.
We find that it is possible to count the number of elements of sets A and L or that they contain a finite number of elements. Such sets are called finite sets.
Now, consider the set B of all even numbers. We cannot count all of them
i.e., we see that the number of elements of this set is not finite.
Similarly , all the elements of J cannot be listed. We find that the number of elements in B and J is infinite. Such sets are called infinite sets.
We can draw many numbers of straight lines passing through a given point. So this set is infinite.
Similarly, it is not possible to find out the last even number or odd number among the collection of all integers. Thus, we can say a set is infinite if it is not finite.
Consider some more examples:
(!) let 'w' be the set of the days of the week. Then W is finite.
(!!) Let 'S' be the set of solutions of the equation x^2 - 16 = 0. Then S if finite.
(!!!) Let 'G' be the set of points on a line. Then G is infinite.
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Example-3. State which of the following sets are finite or infinite.
(!) { x : x ∈ N and (x-1)(x-2) =0}
sol) x can take the values 1 or 2 in the given case. The set is {1,2}. Hence, it is finite.
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(!!) {x : x ∈N and x^2 =4}
sol) x^2 = 4, implies that x = +2 or -2. But x ∈N or x is a natural number so the set is {2}. Hence, it is finite.
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(!!!) { x : x ∈ N and 2x-2 =0}
sol) In a given set x= 1 and 1∈ N. Hence, it is finite.
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(!V) { x : x ∈ N and x is a prime}
sol) The given set is the set of all prime numbers. There are infinitely many prime numbers. Hence, set is infinite.
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(V) { x : x ∈ N and x is odd }
sol) Since there are infinite number of odd numbers, hence the set is infinite.
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Now, consider the following finite sets:
A = {1,2,4}
B = { 6,7,8,9,10}
C = {x :x is a alphabet in the word "INDIA"}
Here,
Number of elements in set A = 3
Number of elements in set B = 5
Number of elements in set C = 4
(In the set C, the element 'I' repeats twice. We know that the elements of a given set should be distinct. So, the number of distinct elements in set C is 4).
The number of elements in a set is called the cardinal number of the set. The cardinal number of the set A is denoted as n(A) = 3.
Similarly, n(B)=5 and n(C)=4.
Note : There are no elements in a null set. The cardinal number of that set is 0.
.^. n(∅) = 0
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Example-4. If A = {1,2,3}; B={a,b,c} then find n(A) and n(B)
sol) The set A contains three distinct elements
.^. n(A) = 3
The set B contains three distinct elements
.^. n(B) = 3
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Exercise - 2.2
1) if A = {1,2,3,4} ; B= {1,2,3,5,6} then find
1) A intersection B and !!) B intersection A
Sol) Are they equal? Yes
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2) If A= { 0,2,4 }, find A intersection null-set and A intersection A
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3) if A = { 2,4,6,8,10 } and B = { 3,6,9,12,15} find A-B and B-A.
1) A-B
Sol) Only the elements which are in A should be taken.
A-B = { 2,4,6,8,10 } -- { 3,6,9,12,15} = { 2,4,8,10}
Since "6" are the only element which is present in both "A" and "B" set
Thus "6"is no taken from "A" set
2) B -- A
Sol) Similarly elements which are only in "B" are taken.
B -- A = { 3,6,9,12,15 } -- { 2,4,6,8,10}
= {3,9,12,15}
NOTE:- A--B not equal to B-A
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4) If A and B are two sets such that A subset of B, then what is A U B
Sol) Let A= {2,3} B = { 2,3,4}
We can see clearly that every element of A is in B.
Thus, A is a subset of B
A U B = { 2,3 } U {2,3,4}
= { 2,3,4}
= B
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5) if A = { x : x is a natural number },
B= { x : x is an even natural number}
C = { x : x is an odd natural number } and
D = { x : x is a prime number }
Sol)
Set A = { x : x is a natural number } ( collection of only natural numbers)
= { 1,2,3,4.....so on }
Set B = { x : x is an even natural number } ( collection of only even natural no's)
= { 2,4,6,8,10.......so on }
Set C = { x : x is an odd natural number }
= {1,3,5,7,9,11,...... so on }
Set D = { x : x is a prime number }
= { 2,3,5,7,11.........so on }
Find:-
1) A n B
=> { Natural no's } n {
=> A { 1,2,3,4,5,6.....} n { 2,4,6.8,10,12......}
=.> {2,4,6,8,10,12....}
=>B
.^. A n B = B
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2) A n C
=> { Natural no's } n { odd natural no's }
=> { 1,2,3,4,5......} n { 1,3,5,7,9...}
=> { 1,3,5,7,9......}
=> C
.^. A n C = C
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3) A n D
=> { Natural numbers } n { Prime numbers }
=> { 1,2,3,4,5......} n { 2,3,5,7,11....}
=> {2,3,,5,7,11}
=> D
.^. A n D = D
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4) B n C
=> { Even natural no} n { odd natural no}
=> { 2,4,6,8,10....} n { 1,3,5,7,9....}
=> { No matching elements }
=> { }
=> empty or null set0
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5) B n D
=> {even natural no} n { Prime number }
=> { 1,2,3,4,5..........} n { 2,3,5,7,11...}
=> { 2 }
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6) C n D
=> { odd natural no} n { prime number }
=> { 1,3,5,7,9....} n { 2,3,5,7,11....}
=> { 3,5,7,11,13....}
=> { odd but prime numbers}
=> { x: x is odd prime number }
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6. If
A = { 3,6,9,12,15,18,21 }
B = { 4,8,12,16,20 }
C= { 2,4,6,8,10}
D = { 5,10,15,20 }
Find
(1) A -- B
=> { 3,6,9,12,15,18,21 } -- { 4,8,12,16,20 }
Removing element "12" from set A which is also present in set B
=> {3,6,9,15,18,21}
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(2) A -- C
=> { 3,6,9,12,15,18,21 } -- { 2,4,6,8,10,12,14,16}
removing {6,12} from set A which is also present in set B
=> { 3,9,15,18,21 }
------------------------------------------------------------------------
(3) A -- D
=> { 3,6,9,12,15,18,21} -- { 5,10,15,20 }
=> { 3,6,9,12,18,21}
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(4) B -- A
=> { 4,8,12,16,20 } -- { 3,6,9,12,15,18,21 }
=> { 4,8,16,20 }
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(5) C -- A
=> { 2,4,6,8,10,12,14,16} -- { 3,6,9,12,15,18,21}
=> { 2,4,8,10,14,16}
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(6) D -- A
=> {5,10,15,20} -- { 3,6,9,12,15,18,21}
=> { 5,10,20}
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(7) B--C
=> { 4,8,12,16,20} --- { 2,4,6,8,10,12,14,16}
=> {20}
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(8) B -- D
=> { 4,8,12,16,20} -- { 5,10,15,20}
=> { 4,8,12,16}
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(9) C-- B
=> { 2,4,6,8,10,12,14,16} -- { 4,8,12,16,20}
=> { 2,6,10,14}
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(10) D -- B
=> { 5,10,15,20} -- { 4,8,12,16,20 }
=> {5,10,15}
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(*) Disjoint Sets :- Suppose A { 1,2,3} and B { 4,5,6}. We see that there are no common elements
in A and B . Such sets are known as disjoint sets.
if A n B =empty set=Null set= void set = { } =0
then the set are disjoint
7) State whether each of the following statement is true or false .justify your answers.
(1) { 2,3,4,5 } and { 3,6} are disjoint sets.
A) False,
=> { 2,3,4,5} n { 3,6 }
=> {3}
Since { 3} is common element in both sets .Thus, its not disjoint sets.
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(2) { a,e,i,o,u} and { a,b,c,d} are disjoint sets.
A) False
=> { a,e,i,o,u } n { {a,b,c,d}
=> { a }
Since { a } is common element in both sets.
Thus, its not disjoint sets
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(3) { 2,6,10,14} and { 3,7,11,15} are disjoint sets.
A) True.
=> { 2,6,10,14} n{ 3,7,11,15}
=> { }
Since there are no common elements . It is an empty set
Thus its disjoint
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(4) { 2,6,10 } and { 3,7,11} are disjoint sets.
A) True.
=> { 2,6,10 } n { 3,7,11}
=> { }
=>0
No common elements. Thus both sets are disjoint.
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(*) Using diagrams to represent sets.
If S is a set and x is an object then either x∈S or x∉S .
Every set can be represented by drawing a closed curve C where elements of C are represented by points within C and elements not in the set by points outside C.
Example: C={1,2,3,4} can be represented as shown below
(*) Universal set and subsets
Let us consider that a cricket team is to be selected from your school. What is the set from which the team can be selected?
It is the set of all students in your school. Now, we want to select the hockey team.
Again, the set from which the team will be selected is the set of all students in your school.
So, for selection of any school team, the students of your school are considered as the universal set.
(!) If we want to study the various groups of people of our state, universal set is the set of all people in Andhra Pradesh.
(!!) If we want to study the various groups of people in our country, universal set is the set of all people in India.
The universal set is denoted by 'μ' .
The Universal set is usually represented by rectangles.
If the set of real numbers R; is the universal set then what about rational and irrational numbers?
Let us consider the set of rational numbers.
Q = {x:x = p , p,q∈ z and q =/= 0}
q
which is read as 'Q' is the set of all numbers such that x equals p/q, where p and q are integers and q is not zero.
Then we know that every element of Q is also an element of R. So, we can say that Q is a subset of R.
If Q is a subset of R, then we write is as Q⊂ R.
Note: It is often convenient to use the symbol '=>' which mean implies.
Using this symbol, we can write the definition of subset as follows:
A ⊂B if a∈ A => a∈ B,
where A,B are two sets.
We read the above statement as "A is a subset of B if 'a' is an element of A implies that 'a' is also an element of B".
Real numbers R; has many subsets.
For example,
The set of natural number N ={1,2,3,4,5...}
The set of whole number W = {0,1,2,3....}
The set of integers Z = {-3,-2,-1,0,1,2,3...}
The set of irrational numbers Q', is composed of all real numbers that are not rational.
Thus, Q' = { x:x∈ R and x∉ Q}
i.e., all real numbers that are not rational.
e.g. √2,√5, and 𝛑
Similarly, the set of natural numbers,
N is a subset of the set of whole numbers W
and we can write N⊂ W.
Also W is a subset of R.
That is N⊂ W and W⊂ R
=>N⊂ W⊂ R
Some of the obvious relations among these subsets are
N⊂Z⊂Q
Q⊂R
Q'⊂R and
N ⊄ Q'
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Example-5. Consider a set of vowels letters,
V= {a,e,i,o,u} . Also consider the set A, of all letters in the English alphabet.
A={a,b,c,d,.....z}. Identify the universal set and the subset in the given example.
sol). We can see that every element of set V is also an element A. But every element of A is not a part of V.
In this case, V is the subset of A.
In other words V⊂ A since whenever a∈V,
then a∈A.
Note: Since the empty set ∅has no elements, we consider that∅ is a subset of every set.
let us consider some more examples of subsets.
> The set C = {1,3,5} is a subset of D={5,4,3,2,1},since each number 1,3, and 5 belonging to C also belongs to D.
> Let A={a,e,i,o,u} and B={a,b,c,d} then A is not a subset of B. Also B is not a subset of A.
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(*) Equal Sets
Consider the following sets.
A = { Sachin, Dravid, Kohli}
B = { Dravid, Sachin, Dhoni}
C = { Kohli, Dravid, Sachin}
We observe that .All the players that are in A are in C but not in B.
Thus, A and C have same elements but some elements of A and B are different.
So, the sets A and C are equal sets but sets A and B are not equal.
Two sets A and C are said to be equal if every element in A belongs to C and every element in C belongs to A.
If A and C are equal sets, than we write A = C.
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Example-6. Consider the following sets :
A = {p,q,r} B={q,p,r}
sol) In the above sets, every element of A is also an element of B
.^. A ⊂ B
Similarly every element of B is also in A.
.^. B⊂ A
Thus, we can also write that if
B⊂ A and A⊂ B <=> A = B.
Here <=> is the symbol for two way implication and is usually read as, if and only if (briefly written as "iff").
*****************************************
Examples-7. If A={1,2,3,....} and N is a set of natural numbers, check whether A and N are equal?
sol) The elements are same in both the sets. Therefore, both A and N are the set of Natural numbers.
Therefore the sets A and N are equal sets or A=N.
*****************************************
Example-8. Consider the sets A ={p,q,r,s} and B={1,2,3,4}. Are they equal?
sol) A and B do not contain the same elements.
So, A=/= B.
*****************************************
Example-9. Let A be the set of prime numbers less than 6 and P the set of prime factors of 30. Check if A and P are equal.
sol) The set of prime numbers less than 6
A={2,3,5}
The prime factors of 30 are 2,3 and 5. So,
P={2,3,5}
Since the elements of A are the same as the elements of P, therefore, A and P are equal.
*****************************************
Example-10. Show that the sets A and B are equal, where
A = {x:x is a letter in the word 'ASSASSINATION'}
B = {x : x is a letter in the word STATION}
sol) Given,
A={x:x is a letter in the word ' ASSASSINATION'}
This set A can also be written as A={A,S,I,N,T,O} since generally elements in a set are not repeated.
Also given that
B = { x : x is a letter in the word STATION}
'B' can also be written as B = { A,S,I,N,T,O}
So, the elements of A and B are same and
A = B
*****************************************
Exercise - 2.3
1) Which of the following sets are equal?
(1) A = { x: x is a letter in the world FOLLOW}
(2) B = { x: x is a letter in the FLOW }
(3) C = { x : x is a letter in the word WOLF }
Sol ) All the above sets are in set-builder form.Lets convert into roster form
Te roster form of set A = { F , O , L , W }
(since same elements in a set cannot be repeated)
B Roster form of Set B = { F, L, O W }
C= Roster form of Set C = { W, O,L.F}
All the three sets A, B and C have same elements { F,L,O,W}.
So the three sets are equal. A=B=C
---------------------------------------------------------------------------
2.) Consider the following sets and fill up the blanks in the statement given below with "=" or
=/= as to make the statement true.
A) = {1,2,3}
B = { the first three natural numbers }
C= { a,b,c,d}
D = { d,c,a,b}
E = { a,e,i,o,u}
F = { set of vowels in English Alphabet }
(1) A ---- B
Sol) { 1,2,3 } ----- { the first three natural numbers }
=> { 1,2,3} ---- { 1,2,3}
=> Set A = Set B
=> A = B
-----------------------------------------------------------------------------
(2) A ---- E
sol) { 1,2,3}------ { a,e,i,o,u}
=> { natural no} ---- { alphabet}
=> Set A =/= Set E
=> A =/= B
------------------------------------------------------------------------------------
(3) C ---- D
Sol) { a,b,c,d} ----- { d,c,a,b}
=> Set C = Set D
=> C = D
--------------------------------------------------------------------------------
(4) D ------ F
Sol) {d,c,a,b} ----- { set of vowels in English Alphabet }
=> {d,c,a,b} ----- { a,e,i,o,u}
Both sets are different
=> Set D =/= Set F
=> D =/= F
--------------------------------------------------------------------------------
(5) F ---- A
Sol ) { set of vowels in English Alphabet } ------ { 1,2,3}
=> {a,e,i,o,u } ----- {1,2,3}
=> Set F =/= Set A
=> F =/= A
-------------------------------------------------------------------------------
(6) D --- E
Sol) { d,c,a,b } ---- { a,e,i,o,u}
=> Set D =/= Set E
=> D =/= E
-------------------------------------------------------------------------------
(7) F ---- B
Sol) { set of vowels in English Alphabet } ---- { The first three natural numbers }
=> { a,e,i,o,u } ---- { 1,2,3}
=> Set F =/= Set B
=> F =/= B
*****************************************************
3) In each of the following, state whether A= B or not.
(1) A = { a,b,c,d } B = { d,c,a,b}
Sol) A = B
Since both the Sets A and B have same elements {a,b,c,d}
----------------------------------------------------------------------------
(2) A = { 4,8,12,16} B = { 8,4,16,18}
Sol) A =/= B
Since not all elements are same in both sets A and B except { 4,8,16}
===================================================
(3) A = { 2,4,6,8,10} B= { x : x is a positive even integer and x<10}
Sol ) A =/= B
Since B { even integer but x < 10 } = { 2,4,6,8,}
All elements are same except {10} in set A
-------------------------------------------------------------------------------------
4) A = { x : x is a multiple of 10 } B = { 10,15,20,25,30}
Sol ) A =/= B
set A { 10,20,30,40.......} B = { 10,15,20,25,30}
Both Sets are different
********************************************************
4) State the reasons for the following.
(1) { 1,2,3......10} =/= { x : x belongs N and 1< x < 10 }
Sol ) Check Set B {x : x belongs N and 1<x<10}
"x belongs to N ( natural no) :- {1,2,3,4....}
" 1<x<10 " :- "x is greater than 1 x>1 but less then "10" x<10
Set A { 1,2,3...10 } =/= Set B { 2(x> 1), 3, 4..... 9 ( x < 10) }
reason :- Since Set A is a set of natural no's starting from "1" and ends at "10" whereas Set B also a set of natural no's but it starts with "2" and ends at "9" ...so both sets are different.
.^. {1,2,3,.....10} =/= { 2,3,4,,,,,9}
---------------------------------------------------------------------------------
(2) { 2,4,6,8,10 } =/= { x : x = 2n+1 and x belongs N}
Sol ) Set B = { x : x = 2n +1 and x belongs N}
2n+1 :-
2(0) +1 = 0+1 = 1
2(1) +1 = 2+1 = 3
2(2) + 1 = 4 +1 = 5
2( 3 ) + 1 = 6+1 = 7
2 ( 4 ) + 1 = 8 +1 = 9
" x= 2n+1" :- ( 1,3,5,7,9....)
Set A { 2,4,6,8,10} are even natural numbers
Set B { 1,3,5,7,.....) are set of odd natural numbers
Set A { 2,4,6,8,10} =/= Set B { 1,3,5,7,9....}
Reason :- Set A is a set of even natural no's whereas Set B is on odd natural no's..both sets are different.
.^. { 2,4,6,8,10} =/= { 1,3,5,7,9....]
-------------------------------------------------------------------------------------
3) { 5,15,30,45 } =/= { x:x is a multiple of 15}
Sol) Set B { x : x is a multiple of 15} = { 15 ,30,45 ,60.....}
Set A { 5,15,30,45 } =/= Set B { 15,30,45,60....}
Reason :- Since {5} does not exists in Set B
.^. {5,15,30,45} =/= { 15,30,45,60}
---------------------------------------------------------------------------------
4) {2,3,5,7,9} =/= { x:x is a prime number }
Sol ) Set B { x ; x is a prime number } = { 2,3,5,7,11...}
Set A { 2,3,5,7,9} =/= Set B { 2,3,5,7,11...]
Reason :- Since 9 is not a prime number which does not exists in set b
.^. {2,3,5,7,9} =/= {2,3,5,7,11...}
*********************************************************
5.) List all the subsets of the following sets.
(1) B = {p,q}
Sol) Subsets with one element { p }, { q}
Subsets with two elements { p,q}
Empty set is also a subset {0 }
Thus the list of all subsets of {p, q} will be.
{p} , {q} , {
p,q}, {0 }. = Total 4 subsets .
Note :- if you want to check whether you found all subsets or not just use
2^ n formula where n= no of elements
example { p , q} = 2 elements
2^2 = 4
------------------------------------------------------------------------------------------
(2) C = { x,y,z }
Sol ) Subsets with one element { x}, { y }, { z }
Subsets of two elements { x,y}, { x, z }, { y, z }
Subsets of two three elements { x,y,z}
Empty set {0 }
Thus, the lists of { x,y,z } all subsets will be
{ x } , { y }, { z },
{ x,y },{x,z}, {y,z }, {
x, y, z}, {0 } = Total 8 subsets
2^3 = 2*2*2=8 elements
-------------------------------------------------------------------------------------
(3) D = { a,b,c,d}
Sol ) Subsets with one element { a }, { b }, { c }, { d }
Subsets with two elements { a,b }, { a,c } , { a,d }, { b , c } , {b,d } , { c, d }
Subsets with three elements { a,b,c } , {a,b,d }, {a,c,d }, {b,c,d }
Subsets with four elements { a,b,c,d }
Null or empty set {0 }
Thus, the lists of { a,b,c,d } all subsets will be.
{ a }, { b } , { c }, {d },
{ a, b }, { a ,c } , { a,d }, b, c } , { b, d } , { c , d},
{ a,b,c }, { a, b,d }, { a, c d }, { b,c,d }, {
a,b,c,d }, {0 } = total 16 subsets.
2^ 4 = 2*2*2*2 = 16 elements
--------------------------------------------------------------------------------
4) E = { 1,4,9,16 }
Sol ) One elements subset { 1 }, { 4 }, { 9 }, { 16 }
Two element subsets { 1,4 } , { 1,9 } , { 1, 16 } , { 4, 9 }, { 4, 16 }, { 9, 16 }
3 elements subsets { 1,4,9 }, { 1,4,16 }, {1,9, 16 }, { 4,9, 16}.
4 elements subsets { 1,4,9,16 }
Null or empty set {0 }
Thus the lists of {1,4,9,16} all subsets will be.
{ 1 }, { 4 }. {9 }, { 16}
{ 1,4 }, { 1,9 }, { 1, 16 }, { 4,9 }, { 4,16 }, { 9,16}
{ 1,4,9 }, { 1,4,16 }, 1,9,16}, {4,9,16}
{ 1,4,9, 16 } , {0 } = total 16 subsets
------------------------------------------------------------------------------------
5) F = { 10, 100, 1000 }
Sol ) 1 element subset { 10 }, { 100 }, { 1000 }
2 element subsets { 10, 100 }, { 10, 1000 }, { 100, 1000 }
3 elements subsets { 10, 100, 1000 }
Null or empty set {0}
List of { 10, 100, 1000 } all subsets is
{ 10 }, { 100 }, { 1000}
{ 10, 100 }, { 10, 1000}, { 100, 1000}
{ 10, 100, 1000}, { } = Total 8 subsets
************************************************************
Consider the set E={2,4,6} and F ={6,2,4}
Note that E=F. Now, since each element of E also belongs to F, therefore E is a subset of F.
But each element of F is also an element of E.
So F is a subset of E.
In this manner it can be shown that every set is a subset of itself.
if A and B contain the same elements, they are equal
i.e., A = B.
By this observation we can say that
"Every set is subset of itself"
*****************************************
Example-11. Consider the sets ∅,
A={1,3}
B = {1,5,9}
C = {1,3,5,7,9}.
Insert the symbol ⊂ or ⊄ between each of the following pair of sets.
(!) ∅.......B
sol) ∅ ⊂ B, as ∅ is a subset of every set.
******************************************
(!!) A......B
sol) A ⊄B, for 3∈ A but 3 ∉ B.
*****************************************
(!!!) A..........C
sol) A⊂ C as 1,3 ∈ A also belong to C.
*****************************************
(!v) B......C
sol) B ⊂ C as each element of B is also an element of C.
******************************************
consider the following sets
A={1,2,3}
B={1,2,3,4}
C={1,2,3,4,5}
All the elements of A are in B
.^. A⊂B.
All the elements of B are in C
.^. B⊂C
All the elements of A are in C
.^. A⊂C
That is ,
A⊂B,
B⊂C => A⊂ C
*****************************************
Exercise-2.4
1). State which of the following statements are true given that,
A={1,2,3,4}
(!) 2∈A
sol) True
******************************************
(!!) 2 ∈ {1,2,3,4}
sol) True
*****************************************
(!!!) A ⊂ {1,2,3,4}
sol) True
******************************************
(!v) {2,3,4} ⊂ {1,2,3,4}
sol) True
*****************************************
(*) State the reasons for the following
(!) {1,2,3....10} =/= {x:x∈N and 1<x<10}
sol) R.H.S condition:
x ∈ N = (1,2,3,4.....)
1<x<10 = (2.....9)
(*) L.H.S : {1,2,3....10}
Since L.H.S does not satisfy the condition on R.H.S as 1 and 10 are excluded.
*****************************************
(!!) {2,4,6,8,10} =/= {x:x=2n+1 and x∈N}
sol) R.H.S condition :
(*) 1st cond:-x = 2n+1
(2(1)+1=3)
(2(2)+1=5)
(2(3)+1=7)
x= 2n+1 => (3,5,7,......)
2nd cond:-x ∈ N => (1,2,3,4...)
(*) L.H.S Condition:
{2,4,6,8,10}
Now,
In the above L.H.S set should be {3,5,7,9...} instead of {2,4,6,8,10}
.^. L.H.S =/= R.H.S
*****************************************
(!!!) {5,15,30,45} =/= {x:x is a multiple of 15}
sol) R.H.S : x is a multiple of 15
ex : 1*15=15, 2*15=30, 3*15=45 so on..
But Here
L.H.S: {5,15,30,45}...5 is not multiple of 15
L.H.S =/= R.H.S
*****************************************
(!V) {2,3,5,7,9} =/= { x:x is a prime number}
sol) Prime number : which divisible by 1 and itself
example:- ( 2,3,5,7,11,13,15.......)
Now,
L.H.S : {2,3,5,7,9.....} here 9 is not prime.
Hence
L.H.S =/= R.H.S
*****************************************
3. List all the subsets of the following sets.
(!) B = {p,q}
sol) No.of elements in set n = 2
No of subsets = 2^n = 2^2 = 4
{ },{p} {q}, {p,q} = 4
Note: Null set { } is the subset of set itself.
******************************************
(!!) C = {x,y,z}
sol) No.of elements n= 3
No. of subsets = 2^3 = 8
Now
{ }, {x},{y},{z},{x,y},{x,z},{y,z},{x,y,z} = 8
******************************************
(!!!) D = {a,b,c,d}
sol) No.of elements n= 4
No.of subsets = 2^4 = 16
{ }, {a}, {b}, {c}, {d}
{a,b}, {a,c}, {a,d},
{b,c}, {b,d},
{c,d}
{a,b,c},{a,b,d},{a,c,d},{b,c,d}
{a,b,c,d}
************
subsets= 16
************
*****************************************
(!v) E = {1,4,9,16}
sol) No. of elements n= 4
No. of subsets = 2^4 = 16
subsets:-
{ }, {1 }, {4}, {9}, {16}
{1,4}, {1,9}, {1,16},{4,9},{4,16},{9,16}
{1,4,9}, {1,4,16}, {1,9,16},{4,9,16}
{1,4,9,16}
******************
Total subsets = 16
*****************
******************************************
(V) F = { 10,100,1000}
sol) No. of elements n = 3
No. Of subsets = 2^3 = 8
Subsets:
{ }, {10}, {100},{1000}
{10,100},{10,1000}, {100,1000}
{10,100,1000}
**************
total subsets = 8
***************
*****************************************
1) State which of the following sets are empty and which are not?
(1) The set of straight lines passing through a point.
Sol) Not empty , Because if a line pass through any point (x,y)
----------------------------------------------------------------------------------
(2) Set of odd natural numbers divisible by 2.
Sol) Empty .
Natural numbers : 1,2,3,4,5,6,7,8,9,10,11.....so.on
odd natural numbers :- 1,3,5,7,9,11.... is any of this odd.no's divisible by "2"?
An odd number is an integer that is not evenly divisible by '2'.
Therefore its is an empty set
----------------------------------------------------------------------------------------
(3) {x:x is a natural number, x<5 and x>7}
Sol) Empty.
natural number x<5 : 1,2,3,4.
natural number x>7 : 8,9,10,11
Is there a number which is less than 5 and greater than 7 simultaneously?NO
Therefore its an empty set.
---------------------------------------------------------------------------------------
(4) {x:x is a common point to any two parallel lines.}
Sol) Empty
Parallel Lines :- Two lines is said to be parallel if an only if they have no common points.
<---------------------------->
<---------------------------->
(parallel lines)
Therefore it is an empty set.
----------------------------------------------------------------------------------------
5) Set of even prime numbers.
Sol) Not empty.
Prime Numbers :- 2,3,5,7,11,........
Even Prime Numbers :- A prime number which is divisible by "2"
We have "2" which is even as well as prime number.
.^. It is not an empty set
========================================================
Finite Set :- if There are only limited number of elements present in a set then such a set is called finite set.
Example :- L= {p,q,r,s} It has only "4" element which is finite
(*) Infinite Set :- if there is an infinite number of elements present in a set. then such a set is called an infinite set.
Example :- A = {x:x is an even number} .
Set A have all even numbers .We cannot count all of them
2. Which of the following sets are finite and infinite.
(1) The sets of months in a year.
Sol) There are only 12 months in a year.. This set has only 12 elements.
.^. the set is finite.
(2) {1,2,3,.....99,100 }
sol) Clearly finite.
(3) The set of prime numbers less than 99.
Sol) Finite.
----------------------------------------------------------------------------------
3) State whether each of the following set is finite or infinite.
(1) The set of letters in the English alphabet.
Sol) Finite. (A,B,C..........Z)
(2) The set of lines which are parallel to the X-axis.
Sol) Infinite.
(3) The set of numbers which are multiples of 5.
Sol) Infinite .
.( 5*1=5...5*2=10 ....5*3=15..........so on)
(4) The set of circles passing through the origin (0,0)
Sol) Infinite. ( infinite number of circles can pass through origin).
*************************************************************
(*) Venn Diagrams
Venn-Euler diagram or simply Venn-diagram is a way of representing relationship between sets. These diagrams consists of rectangles and closed curves usually circles.
The universal set is usually represented by a rectangle.
(!) Consider that μ ={ 1,2,3,.....10} is the universal set of which A = { 2,4,6,8,10} is a subset. Then the venn-diagram is as:
(!!) μ = { 1,2,3,.....10} is the universal set of which , A ={2,4,6,8,10} and B={4,6} are subsets and also B ⊂ A. Then we have the following figure.
(!!!) Let A ={a,b,c,d} and B={c,d,e,f}. Then we illustrate these sets with a Venn diagram as.
(*) Basic operations on sets
We know that arithmetic has operations of additions, subtraction and multiplication of numbers.
Similarly in sets, we define the operation of union, intersection and difference of sets
(*) Union of sets
Example-12. Suppose A is the set of students in your class who were absent on Tuesday and B the set of students who were absent on Wednesday. Then,
A={Roja, Ramu, Ravi} and
B = {Ramu, Preethi, Haneef}
Now, we want to find K, the set of students who were absent on either Tuesday or Wednesday.
Then, does
Roja ∈ K?
Ramu ∈ K?
Ravi ∈ K?
Haneef ∈ K?
Preeti ∈ K?
Akhila ∈ K?
Roja, Ramu, Ravi, Haneef and Preeti all belong to K but Ganpati does not.
Hence,
K = {Roja, Ramu, Raheem, Prudhvi,Preethi}
Here K is called the union of sets A and B. The union of A and B is the set which consists of all the elements of A and B and the common elements being taken only once. The symbol ⋃ is used to denote the union.
Symbolically, we write A ⋃ B and usually read as 'A union B'
A ⋃ B = { x : x∈A or x∈B }
*****************************************
Example-13. let A= {2,5,6,8} and B={5,7,9,1}. Find A⋃ B
sol) A ⋃ B = { 1,2,5,6,7,8,9}
Note: common element 5 was taken only once while writing A ⋃ B
*****************************************
Example-14. let A= {a,e,i,o,u} and B={a,i,u}. Show that A ⋃ B = A.
This example illustrate that union of sets A and its subset B is the Set A itself.
i.e, if B ⊂A, then A⋃ B = A.
The union of the sets can be represented by a Venn-diagram as shown(shaded portion)
*****************************************
Example-15. Illustrate A⋃ B in Venn-diagram where.
A= {1,2,3,4} and B = {2,4,6,8}
sol) A ⋃ B = { 1,2,3,4,6,8}
(*) intersection of sets.
Let us again consider the example of absent students. This time we want to find the set L of students who were absent on both Tuesday and Wednesday.
We find that L = { Ramu}.
Here, L is called the intersection of sets A and B.
In general, the intersection of sets A and B is the set of all elements which are common to A and B.i.e., those elements which belongs to A and also belong to B.
We denote intersection by A∩B. ( read as "A intersection B"). Symbolically , we write.
A∩B = { x : x∈A and x∈B}
The intersection of A and B can illustrated in the Venn-diagram as shown in the shaded portion in the adjacent figure.
*****************************************
Example-16. Find A∩ B when A={5,6,7,8} and B={3,4,5}
sol) The common elements in both A and B are 7,8.
.^. A ∩ B = { 7,8}
*****************************************
Example-17. Illustrate A∩B in Venn-diagram where A={1,2,3} and B = {3,4,5}
sol)
.^. A ∩ B = {3}
*****************************************
(*) Disjoint Set
Suppose A={1,3,5,7} and B={2,4,6,8} .
We see that there are no common elements in A and B. Such sets are known as disjoint sets.
The disjoint sets can be represented by means of the Venn-diagram as follows.
A ∩ B = ∅
*****************************************
(*) Difference of Sets
The difference of sets A and B is the set of elements which belong A but do not belong to B.
We denote the difference of A and B by A-B or simply " A minus B".
A- B = { x : x ∈ A and x ∈ B}
*****************************************
Example-18. Let
A = { 1,2,3,4,5,6},
B= { 4,5,6,7}
Find A - B
sol) Given A ={ 1,2,3,4,5} and B={4,5,6,7} . Only the elements which are in A but not in B should be taken.
A - B = {1,2,3}
Since 4,5 are the elements in B are not taken.
Similarly for B-A, the elements which are only in B are taken.
.^. B - A = { 6,7}
(4,5 are the elements in A)
Note that A - B =/= B - A
The Venn-diagram of A - B is as shown
Example-19. Observe the following
A = {3,4,5,6,7}
.^. n(A) = 5
B = {1,6,7,8,9}
.^. n(B) = 5
A⋃ B = { 1,3,4,5,6,7,8,9}
.^. n(A⋃ B) = 8
A ⋃B = { 6,7}
.^. n(A⋃B) = 2
.^. n(A ⋃ B) = 5 + 5 - 2 = 8
We observe that
n(A⋃ B) = n(A) + n(B) - n (A B)
*****************************************
Exercise- 2.5
1). If A={1,2,3,4}; B={1,2,3,5,6} then find A∩B and B∩A . Are they equal?
sol) A∩B = { 1,2,3,4}∩ { 1,2,3,5,6}
common terms in both sets
.^. A ∩ B = { 1,2,3}
*****************************************
B ∩ A = {1,2,3,5,6} ∩ {1,2,3,4}
B ∩ A = {1,2,3}
A ∩ B and B ∩ A are equal.
******************************************
2). A={0,2,4}, find A ∩∅ and A∩A. Comment.
sol). A ∩∅={0,2,4} ∩ { }
=> A ∩ ∅ = { }
******************************************
A∩A = { 0,2,4} ∩ {0,2,4}
=> {0,2,4}
*****************************************
3). If A={2,4,6,8,10} and B={3,6,9,12,15},
find A-B and B-A.
sol)A - B = {2,4,6,8,10} - {3,6,9,12,15}
=> {2,4,8,10}
B-A={3,6,9,12,15} - {2,4,6,8,10}
=> {3,9,12,15}
*****************************************
4). If A and B are two sets such that A⊂B then what is A ∪B?
sol) Let
A = {1,2,3,4}
B = {1,2,3,4,5,6}
A⊂ B (Every element of A is in B)
Now,
A ⋃ B = {1,2,3,4} ⋃ {1,2,3,4,5,6}
=> { 1,2,3,4,5,6} = B
*****************************************
5). If A={x : x is a natural number }
B={x:x is an even natural number}
C={x:x is an odd natural number}
D={x :x is a prime number}
find
first, let's write all the sets in roster form.
A={x:x is natural number}
= {1,2,3,4,5,6,7...}
B={x : x is an even natural number}
= {2,4,6,8,10,12,14.....}
C = { x : x is an odd natural number}
= { 1,3,5,7,9....}
D = {x : x is prime number}
= { 2,3,5,7,11,13,15,17.....}
*****************************************
(!) A∩B
sol) Common elements in A & B
{natural numbers} ∩ {even natural numbers}
{1,2,3,4,5,6,7...} ∩ {2,4,6,8,10,12,..}
{2,4,6,8,.......} = B (even natural number)
******************************************
(!!) A∩C
sol) Common elements between A & C
{natural number} ∩ { odd natural number}
{ 1,2,3,4,5,6,7...} ∩ {1,3,5,7,9,11,13,..}
{1,3,5,7,...} = C (odd natural number)
*****************************************
(!!!) A∩D
sol) common elements in A & D
{natural number} ∩{prime number}
{1,2,3,4,5,6,7...} ∩ {2,3,5,7,11,13,15,17..}
{2,3,5,7....} = D (prime number)
*************************************** (!V) B∩C,
sol) Common elements in B&C
{even natural number}∩{odd natural number}
{ 2,4,6,8,10,12,14...} ∩ { 1,3,5,7,9...}
{ } = ∅ (empty set)
******************************************
(V) B∩D
sol) {even natural number}∩{prime number}
{2,4,6,8,10....} ∩ {2,3,5,7,11...}
{ 2 }
*****************************************
(V!) C∩D.
sol) {odd natural number} ∩ { prime number}
{ 1,3,5,7,9,11,13...} ∩ { 2,3,5,7,11,13...}
{ 3,5,7,11,13....}
******************************************
6). If
A={3,6,9,12,15,18,21}
B={4,8,12,16,20}
C= {2,4,6,8,10,12,14,16}
D={5,10,15,20}
find
(!) A - B
sol) {3,6,9,12,15,18,21}- {4,8,12,16,20}
= { 3,6,9,15,18,21}
****************************************
(!!) A - C
sol) {3,6,9,12,15,18,21} - {2,4,6,8,10,12,14,16}
A - C = {3,9,15,18,21}
*****************************************
(!!!) A - D
sol) {3,6,9,12,15,18,21}-{5,10,15,20}
A - D = {3,6,9,12,18,21}
*****************************************
(!v) B - A
sol) {4,8,12,16,20} - {3,6,9,12,15,18,21}
B - A = { 4,8,16,20}
****************************************
(V) C - A
sol) { 2,4,6,8,10,12,14,16} - {3,6,9,12,15,18,21}
C - A ={2,4,8,10,14,16}
******************************************
(V!) D - A
sol) {5,10,15,20} - {3,6,9,12,15,18,21}
D - A = {5,10,20}
****************************************
(V!!) B- C
sol) {4,8,12,16,20} - {2,4,6,8,10,12,14,16}
B - C = {20}
*****************************************
(V!!!) B-D
sol) {4,8,12,16,20} - { 5,10,15,20}
B - D = {4,8,12,16}
*****************************************
(!x) C - B
sol) {2,4,6,8,10,12,14,16} - { 4,8,12,16,20}
C - B = {2,6,10,14}
*****************************************
(x) D - B
sol) {5,10,15,20} - { 4,8,12,16,20}
D - B = {5,10,15}
*****************************************
7. State whether each of the following statement is true or false. Justify your answers.
(!) { 2,3,4,5} and {3, 6} are disjoint sets.
sol) Two sets are disjoint if they have no common element.
We know :- intersection of two sets means common element in both sets
Now,
{2,3,4,5} ∩ {3,6}
= { 3} ≠ ∅
Since, there is a common element in both set.
The given sets are not disjoint.
So, the given statement is False.
*****************************************
(!!) {a,e,i,o,u} and {a,b,c,d} are disjoint sets.
sol) {a,e,i,o,u} ∩ {a,b,c,d}
{a} ≠ ∅
Since, there is a common element in both set.
The given sets are not disjoint.
So, the given statement is False.
*****************************************
(!!!) {2,6,10,14} and {3,7,11,15} are disjoint sets.
sol) No common elements in both sets
Hence ,its disjoint
So, the given statement is True
*****************************************
(!V) { 2,6,10} and {3,7,11} are disjoint sets.
sol) no common elements in both sets
Hence pair is disjoint
So, the given statement is True.
*****************************************
---------------------------------------------------------------------------------------
Observe the example given below :
1) Euclid, Pythagoras, Gauss, Leibnitz, Aryabhatta, Bhaskar.
2) a,e,i,o,u
3) Happy, sad, angry, anxious, joyful, confused.
4) Cricket, football, kho-kho, kabaddi, basketball
5) 1,3,5,7,9......
What do you observe?
Example 1 is a collection of name of some mathematicians.
Example 2 is a collection of vowel letters in the English alphabet
Example 3 is a collection of feelings.
We see that the name/items/objects in each example have something in common.
i.e., they form a collection .
We come across collections in mathematics too.
For example:
> natural numbers,
>prime numbers,
>quadrilateral in a plane etc.
All examples seen so far are well defined collection of objects or ideas.
A well defined collection of objects or ideas is known as a Set.
(*) Well defined Sets
What do we mean when we say that a set is a well defined collection of objects. Well defined means that :
1) All the objects in the set should have a common feature or property; and
2) It should be possible to decide whether any given object belongs to the set or not.
Let us understand ' well defined' through some examples.
Consider the statement : The collection of all tall students in your class.
What difficulty is caused by this statement? Here, who is tall is not clear.
*) Richa decides that all students taller than her are tall. Her set has five students.
*) Yashodhara also decides that tall means all students taller than her. her set has ten students.
*) Ganapati decides that tall means every student whose height is more than 5 feet. His set has 3 students.
We find that different people get different collections. So, this collection is not well defined.
Now consider the following statement : The collection of all students in your class who are taller than 5 feet 6 inches.
In this case, Richa, Yashodhara and Ganapati, all will get the same collection . So, the collection forms a well defined set.
(*) Naming of Sets and Elements of a Set
We usually denote a set by upper case letters, A,B,C,X,Y,Z etc. A few examples of sets in mathematics are given below.
*) The set of all Natural numbers is denoted by N.
*) The set of all Integers is denoted by Z.
*) The set of all Rational numbers is denoted by Q.
*) The set of all Real numbers is denoted by R.
Notice that all the sets given above are well defined collections because given a number we can decide whether it belongs to the set or not.
Let us see some more examples of elements.
Suppose we define a set as all days in a week, whose name begins with T.
Then we know that Tuesday and Thursday are part of the set but Monday is not.
We say that Tuesday and Thursday are elements of the set of all days in a week starting with T.
Consider some more examples :
(!) We know that N usually stands for the set of all natural numbers. Then 1,2,3.....are elements of the set. But 0 is not an element of N.
(!!) Let us consider the set B, of quadrilaterals.
B={ square, rectangle, rhombus, parallelogram}
Can we put triangle, trapezium or cone in the above set 'B'?
No, a triangle and cone are cannot be members of B.
But a trapezium can be a member of the set B.
So, we can say that an object belonging to a set is knows as a member/ element of the set. We
use the symbol ∈ to denote 'belongs to'.
So 1∈ N means that 1 belongs to N. Similarly 0 ∌ N means that 0 does not belong to N.
There are various ways in which we can write sets.
For Example, we have the set of all vowel letters in the English alphabet. Then, we can write :
(!) V = {a,e,i,o,u}, Here,we list down all the elements of the set between chain/curly brackets. This is called the roster form of writing sets.
In roster form, all elements of the set are written, separated by commas, within curly brackets.
(!!) V= { x:x is a vowel letter in the English alphabet}
or V = {x/x is a vowel letter in the English alphabet}
This way of writing a set is known as the set builder form.
Here , we use symbol x(or any other symbol y,z etc.,) for the element of the set. This is followed by a colon (or a vertical line) , after which we write the characteristic property possessed by the elements of the set.
The whole is enclosed within curly brackets.
Let C = { 2,3,5,7,11}, a set of prime numbers less than 13. this set can be denoted as :
C = {x/x is a prime number less than 13} or
C = {x:x is a prime number less than 13}.
*****************************************
Example-1. Write the following in roster and set builder form.
(!) The set of all natural numbers which divide 42.
sol) Let B be the set of all natural numbers which divide 42. Then, we can write :
B = { 1,2,3,6,7,14,21,42}= Roster form
B ={x:x is a natural number which divides 42} = Set builder form
*****************************************
(!!) The set of natural numbers which are less than 10.
Sol) let A be the set of all natural numbers which are less than 10. Then, we can write :
A={1,2,3,4,5,6,7,8,9}= Roster form
B={x:x is a natural number which is less than 10} = set builder form
*****************************************
Note :
(!) In roster form, the order in which the elements are listed is immaterial. Thus, in example 1, we can also write {1,3,7,21,2,6,4,42}
(!!) While writing the elements of a set in roster form, an element is not repeated. For example, the set of letters forming the word "SCHOOL" is {S,C,H,O,,L} and not {S,C,H,O,O,L}
*****************************************
Example-2, Write the set B={x:x is a natural number and x^2<40 } in the roster form.
sol) We look at natural numbers and their squares starting from 1. When we reach 7, the square is 49 which is greater than 40. The required numbers are 1,2,3,4,5,6.
So, the given set in the roster form is
B = { 1,2,3,4,5,6}.
*****************************************
Exercise-2.1
1) Which of the following are sets? Justify your answer?
(1) The collection of all the months of a year beginning with the letter "J".
Sol) This is a set .why?
Because the months starts with a letter "j" are ( January,June,July) ..
We also know no extra month can be added or removed or replaced from year calendar .
Thus , it is well defined set.
----------------------------------------------------------------------------------------------
(2) The collection of ten most talented writers of India.
sol) This is not a set. why?
Because the top ten writers names today can be replace by other writers name in future.
Thus , it is not well-defined set.
---------------------------------------------------------------------------------------------
(3) A team of 11 best cricket batsmen of the world.
Sol) The names of 11 best batsmen can replace by other batsman in future.
Thus it is not a set.
----------------------------------------------------------------------------------------------
(4) The collection of all boys in your class.
Sol) This is a set
--------------------------------------------------------------------------------------------
(5) The collection of all even integers.
Sol) This is a set.
----------------------------------------------------------------------------------------------------------------------------
2) If A = {0,2,4,6 }, B= {3,5,7} and C= {p,q,r} then fill the appropriate symbols,
1) 0.....∈......A (Since 0 is an element of A)
2) 3....∉.......C (Since 3 is not an element of C)
3) 4.....∉......B (Since 4 is not an element in B)
4) 8.....∉......A (Since 8 is not an element in A)
5) p....∈.......C (Since p is an element in C)
6) 7....∈.......B (Since 7 is an element in B)
******************************************
3) Express the following statements using symbols.
(1) The elements X does not belong to "A"
sol) x ∉ A
*************************************
(2) "d" is an element of the set "B"
sol) d ∈ B
*****************************************
(3) '1" belongs to the set of Natural numbers N.
sol) 1 ∈ N
******************************************
(4) "8" does not belong to the set of prime numbers "P"
sol) 8 ∉ P
******************************************
4) State whether the following statements are true of false.
(1) 5 does not belongs to { prime numbers }
Sol) False . As we know "5' can only be divided by "1" and "itself" .So it is a prime numbers.
******************************************
(2) S = {5,6,7 } implies 8 belongs to S
Sol) False.
******************************************
(3) -5 does not belongs to "W" where "W" is the set of whole numbers.
Sol) True. As we know whole number does not contain fractional, or decimal or negative numbers..It contain only positive integers.(0,1,2,3,...)
*****************************************
(4) 8/11 belongs to Z where "Z" is the set of integers.
Sol) False . An integer does not contain fractional part.
***********************************************
5. Write the following sets in roster form.
(1) B = { x : x is a natural number less than 6 }
Sol) B = {1,2,3,4,5}
*****************************************
(2) C = { x : x is a two-digit natural number such that the sum of its digits is 8}
Sol) C = {17,26,35,44,53,62,71}
((17) 1+7=8,..(26)2+6=8,...so on)
******************************************
(3) D = {x : x is a prime number which is a divisor of 60 }
Sol) D = {2,,3,5 }
(60/2=30......60/5 =12...60/3=20...)
*****************************************
(4) E = {the set of all letters in the word BETTER }
Sol) E = {B,E,T,R}
*****************************************
6) Write the following sets in the set-builder form.
(1) {3,6,9,12 }
Sol) if we observe the set we can clearly see it is in multiple of "3"
= ( 3*1=3.....3 *2= 6,...3*3=9.....3*4=12)
={3,6,9,12}
Set-builder:-
A = { x :x is a multiple of 3 and less than 13 }
*****************************************
(2) { 2,4,8,16,32 }
Sol ) it is in power of 2^x form
= (2^1=2....2^2=4...2^3=8.....2^4=16....2^5=32)=
{2,4,8,16,32}
Set-builder :
A = { x : x is in power of 2^x and x is less than 6}
******************************************
(3) { 5,25,125,625}
Sol) It is in 5^x form
= ( 5^1 = 5.....5^2=25.....5^3= 125...5^4=625}=
{5,25,125,625}
Set-builder :-
A= { x :x is in power of 5^x and x is less than 5}
******************************************
(4) {1,4,9,16,25,.....100}
Sol) (1^2=1....2^2=4....3^2=9....4^2=16....5^2=25.....10^2=100)
Set-builder:
A = { x :x is a square of natural number but not greater than 10}
*****************************************
7) List all the elements of the following sets in roster form.
(1) A= { x: x is a natural number greater than 50 but less than 100}
Sol) A = { 51,52,53,54,55..............97,98,99}
******************************************
(2) B = { x: x is an integer, x^2 =4 }
Sol) B= {-2 , 2}
******************************************
(3) D = { x: x is a letter in the word "LOYAL" }
sol) D = { L,O,Y,A}
*****************************************
8) Match the roster form with set-builder form.
(1) {1,2,3,6}
(a) {x:x is prime number and a divisor of 6}
****************************************
(2) {2,3}
(b) {x:x is an odd natural number smallest than 10}
*****************************************
(3) {M,A,T,H,E,I,C,S}
(c){x:x is a natural number and divisor of 6}
*****************************************
(4) {1,3,5,7,9}
(d) {x:x is a letter of the word MATHEMATICS }
SOLUTION:-
1. { 1,2,3,6 } => { x: x is a natural number and divisor of 6}
2. {2,3} => { x:x is prime number and a divisor of 6}
3. {M,A,T,H,E,I,C,S} => { x:x is a letter of the word MATHEMATICS}
4. {1,3,5,7,9} => {x:x is an odd natural number less than 10}
*********************************************************************************
(*) Types of Set
Let us consider the following examples of sets :
(!) A = { x:x is natural number smaller than 1}
(!!) D = {x:x is a odd prime number divisible by 2}
How many elements are there in A and D? we find that there is no natural number which is smaller than 1. So set A contains no elements or we say that A is an empty set.
Similarly, there are no prime numbers that are divisible by 2 . So, D is also an empty set.
A set which does not contain any element is called an empty set, or a Null set, or a void set. Empty set is denoted by the symbol ∅or { }
Here are some more examples of empty sets.
(!) A = { x:1<x<2, x is a natural number}
(!!) B = {x:x^2=0 and x is a rational number}
(!!!) D = {x:x^2 = 4, x is odd }
Note : ∅ and {0} are two different sets. {0} is a set containing the single element 0 while { } is null set.
*****************************************
Finite & Infinite sets
Now consider the following sets :
(!) A = { the students of your school}
(!!) L = {p,q,r,s}
(!!!) B = {x :x is an even number}
(!v) J = { x : x is a multiple of 7 }
In (!), the number of elements will be the number of students in your school.
In (!!) the number of elements in set L is 4.
We find that it is possible to count the number of elements of sets A and L or that they contain a finite number of elements. Such sets are called finite sets.
Now, consider the set B of all even numbers. We cannot count all of them
i.e., we see that the number of elements of this set is not finite.
Similarly , all the elements of J cannot be listed. We find that the number of elements in B and J is infinite. Such sets are called infinite sets.
We can draw many numbers of straight lines passing through a given point. So this set is infinite.
Similarly, it is not possible to find out the last even number or odd number among the collection of all integers. Thus, we can say a set is infinite if it is not finite.
Consider some more examples:
(!) let 'w' be the set of the days of the week. Then W is finite.
(!!) Let 'S' be the set of solutions of the equation x^2 - 16 = 0. Then S if finite.
(!!!) Let 'G' be the set of points on a line. Then G is infinite.
******************************************
Example-3. State which of the following sets are finite or infinite.
(!) { x : x ∈ N and (x-1)(x-2) =0}
sol) x can take the values 1 or 2 in the given case. The set is {1,2}. Hence, it is finite.
*************************************
(!!) {x : x ∈N and x^2 =4}
sol) x^2 = 4, implies that x = +2 or -2. But x ∈N or x is a natural number so the set is {2}. Hence, it is finite.
*************************************
(!!!) { x : x ∈ N and 2x-2 =0}
sol) In a given set x= 1 and 1∈ N. Hence, it is finite.
*************************************
(!V) { x : x ∈ N and x is a prime}
sol) The given set is the set of all prime numbers. There are infinitely many prime numbers. Hence, set is infinite.
*************************************
(V) { x : x ∈ N and x is odd }
sol) Since there are infinite number of odd numbers, hence the set is infinite.
*************************************
Now, consider the following finite sets:
A = {1,2,4}
B = { 6,7,8,9,10}
C = {x :x is a alphabet in the word "INDIA"}
Here,
Number of elements in set A = 3
Number of elements in set B = 5
Number of elements in set C = 4
(In the set C, the element 'I' repeats twice. We know that the elements of a given set should be distinct. So, the number of distinct elements in set C is 4).
The number of elements in a set is called the cardinal number of the set. The cardinal number of the set A is denoted as n(A) = 3.
Similarly, n(B)=5 and n(C)=4.
Note : There are no elements in a null set. The cardinal number of that set is 0.
.^. n(∅) = 0
******************************************
Example-4. If A = {1,2,3}; B={a,b,c} then find n(A) and n(B)
sol) The set A contains three distinct elements
.^. n(A) = 3
The set B contains three distinct elements
.^. n(B) = 3
*****************************************
Exercise - 2.2
1) if A = {1,2,3,4} ; B= {1,2,3,5,6} then find
1) A intersection B and !!) B intersection A
Sol) Are they equal? Yes
----------------------------------------------------------------------------------------------
2) If A= { 0,2,4 }, find A intersection null-set and A intersection A
-------------------------------------------------------------------------------------
3) if A = { 2,4,6,8,10 } and B = { 3,6,9,12,15} find A-B and B-A.
1) A-B
Sol) Only the elements which are in A should be taken.
A-B = { 2,4,6,8,10 } -- { 3,6,9,12,15} = { 2,4,8,10}
Since "6" are the only element which is present in both "A" and "B" set
Thus "6"is no taken from "A" set
2) B -- A
Sol) Similarly elements which are only in "B" are taken.
B -- A = { 3,6,9,12,15 } -- { 2,4,6,8,10}
= {3,9,12,15}
NOTE:- A--B not equal to B-A
*****************************************
4) If A and B are two sets such that A subset of B, then what is A U B
Sol) Let A= {2,3} B = { 2,3,4}
We can see clearly that every element of A is in B.
Thus, A is a subset of B
A U B = { 2,3 } U {2,3,4}
= { 2,3,4}
= B
********************************************************
5) if A = { x : x is a natural number },
B= { x : x is an even natural number}
C = { x : x is an odd natural number } and
D = { x : x is a prime number }
Sol)
Set A = { x : x is a natural number } ( collection of only natural numbers)
= { 1,2,3,4.....so on }
Set B = { x : x is an even natural number } ( collection of only even natural no's)
= { 2,4,6,8,10.......so on }
Set C = { x : x is an odd natural number }
= {1,3,5,7,9,11,...... so on }
Set D = { x : x is a prime number }
= { 2,3,5,7,11.........so on }
Find:-
1) A n B
=> { Natural no's } n {
=> A { 1,2,3,4,5,6.....} n { 2,4,6.8,10,12......}
=.> {2,4,6,8,10,12....}
=>B
.^. A n B = B
-----------------------------------------------------------------------------------
2) A n C
=> { Natural no's } n { odd natural no's }
=> { 1,2,3,4,5......} n { 1,3,5,7,9...}
=> { 1,3,5,7,9......}
=> C
.^. A n C = C
-------------------------------------------------------------------------------
3) A n D
=> { Natural numbers } n { Prime numbers }
=> { 1,2,3,4,5......} n { 2,3,5,7,11....}
=> {2,3,,5,7,11}
=> D
.^. A n D = D
-----------------------------------------------------------------------------------
4) B n C
=> { Even natural no} n { odd natural no}
=> { 2,4,6,8,10....} n { 1,3,5,7,9....}
=> { No matching elements }
=> { }
=> empty or null set
------------------------------------------------------------------------------
5) B n D
=> {even natural no} n { Prime number }
=> { 1,2,3,4,5..........} n { 2,3,5,7,11...}
=> { 2 }
---------------------------------------------------------------------------------
6) C n D
=> { odd natural no} n { prime number }
=> { 1,3,5,7,9....} n { 2,3,5,7,11....}
=> { 3,5,7,11,13....}
=> { odd but prime numbers}
=> { x: x is odd prime number }
****************************************************
6. If
A = { 3,6,9,12,15,18,21 }
B = { 4,8,12,16,20 }
C= { 2,4,6,8,10}
D = { 5,10,15,20 }
Find
(1) A -- B
=> { 3,6,9,12,15,18,21 } -- { 4,8,12,16,20 }
Removing element "12" from set A which is also present in set B
=> {3,6,9,15,18,21}
-----------------------------------------------------------------------
(2) A -- C
=> { 3,6,9,12,15,18,21 } -- { 2,4,6,8,10,12,14,16}
removing {6,12} from set A which is also present in set B
=> { 3,9,15,18,21 }
------------------------------------------------------------------------
(3) A -- D
=> { 3,6,9,12,15,18,21} -- { 5,10,15,20 }
=> { 3,6,9,12,18,21}
----------------------------------------------------------------------
(4) B -- A
=> { 4,8,12,16,20 } -- { 3,6,9,12,15,18,21 }
=> { 4,8,16,20 }
----------------------------------------------------------------------
(5) C -- A
=> { 2,4,6,8,10,12,14,16} -- { 3,6,9,12,15,18,21}
=> { 2,4,8,10,14,16}
------------------------------------------------------------------------------
(6) D -- A
=> {5,10,15,20} -- { 3,6,9,12,15,18,21}
=> { 5,10,20}
-------------------------------------------------------------------------
(7) B--C
=> { 4,8,12,16,20} --- { 2,4,6,8,10,12,14,16}
=> {20}
-------------------------------------------------------------------------
(8) B -- D
=> { 4,8,12,16,20} -- { 5,10,15,20}
=> { 4,8,12,16}
-------------------------------------------------------------------------
(9) C-- B
=> { 2,4,6,8,10,12,14,16} -- { 4,8,12,16,20}
=> { 2,6,10,14}
--------------------------------------------------------------------------
(10) D -- B
=> { 5,10,15,20} -- { 4,8,12,16,20 }
=> {5,10,15}
*****************************************************
(*) Disjoint Sets :- Suppose A { 1,2,3} and B { 4,5,6}. We see that there are no common elements
in A and B . Such sets are known as disjoint sets.
if A n B =empty set=Null set= void set = { } =
7) State whether each of the following statement is true or false .justify your answers.
(1) { 2,3,4,5 } and { 3,6} are disjoint sets.
A) False,
=> { 2,3,4,5} n { 3,6 }
=> {3}
Since { 3} is common element in both sets .Thus, its not disjoint sets.
----------------------------------------------------------------------------------------------
(2) { a,e,i,o,u} and { a,b,c,d} are disjoint sets.
A) False
=> { a,e,i,o,u } n { {a,b,c,d}
=> { a }
Since { a } is common element in both sets.
Thus, its not disjoint sets
------------------------------------------------------------------------------
(3) { 2,6,10,14} and { 3,7,11,15} are disjoint sets.
A) True.
=> { 2,6,10,14} n{ 3,7,11,15}
=> { }
Since there are no common elements . It is an empty set
Thus its disjoint
-------------------------------------------------------------------------
(4) { 2,6,10 } and { 3,7,11} are disjoint sets.
A) True.
=> { 2,6,10 } n { 3,7,11}
=> { }
=>
No common elements. Thus both sets are disjoint.
****************************************
(*) Using diagrams to represent sets.
If S is a set and x is an object then either x∈S or x∉S .
Every set can be represented by drawing a closed curve C where elements of C are represented by points within C and elements not in the set by points outside C.
Example: C={1,2,3,4} can be represented as shown below
(*) Universal set and subsets
Let us consider that a cricket team is to be selected from your school. What is the set from which the team can be selected?
It is the set of all students in your school. Now, we want to select the hockey team.
Again, the set from which the team will be selected is the set of all students in your school.
So, for selection of any school team, the students of your school are considered as the universal set.
(!) If we want to study the various groups of people of our state, universal set is the set of all people in Andhra Pradesh.
(!!) If we want to study the various groups of people in our country, universal set is the set of all people in India.
The universal set is denoted by 'μ' .
The Universal set is usually represented by rectangles.
If the set of real numbers R; is the universal set then what about rational and irrational numbers?
Let us consider the set of rational numbers.
Q = {x:x = p , p,q∈ z and q =/= 0}
q
which is read as 'Q' is the set of all numbers such that x equals p/q, where p and q are integers and q is not zero.
Then we know that every element of Q is also an element of R. So, we can say that Q is a subset of R.
If Q is a subset of R, then we write is as Q⊂ R.
Note: It is often convenient to use the symbol '=>' which mean implies.
Using this symbol, we can write the definition of subset as follows:
A ⊂B if a∈ A => a∈ B,
where A,B are two sets.
We read the above statement as "A is a subset of B if 'a' is an element of A implies that 'a' is also an element of B".
Real numbers R; has many subsets.
For example,
The set of natural number N ={1,2,3,4,5...}
The set of whole number W = {0,1,2,3....}
The set of integers Z = {-3,-2,-1,0,1,2,3...}
The set of irrational numbers Q', is composed of all real numbers that are not rational.
Thus, Q' = { x:x∈ R and x∉ Q}
i.e., all real numbers that are not rational.
e.g. √2,√5, and 𝛑
Similarly, the set of natural numbers,
N is a subset of the set of whole numbers W
and we can write N⊂ W.
Also W is a subset of R.
That is N⊂ W and W⊂ R
=>N⊂ W⊂ R
Some of the obvious relations among these subsets are
N⊂Z⊂Q
Q⊂R
Q'⊂R and
N ⊄ Q'
*****************************************
Example-5. Consider a set of vowels letters,
V= {a,e,i,o,u} . Also consider the set A, of all letters in the English alphabet.
A={a,b,c,d,.....z}. Identify the universal set and the subset in the given example.
sol). We can see that every element of set V is also an element A. But every element of A is not a part of V.
In this case, V is the subset of A.
In other words V⊂ A since whenever a∈V,
then a∈A.
Note: Since the empty set ∅has no elements, we consider that∅ is a subset of every set.
let us consider some more examples of subsets.
> The set C = {1,3,5} is a subset of D={5,4,3,2,1},since each number 1,3, and 5 belonging to C also belongs to D.
> Let A={a,e,i,o,u} and B={a,b,c,d} then A is not a subset of B. Also B is not a subset of A.
*****************************************
(*) Equal Sets
Consider the following sets.
A = { Sachin, Dravid, Kohli}
B = { Dravid, Sachin, Dhoni}
C = { Kohli, Dravid, Sachin}
We observe that .All the players that are in A are in C but not in B.
Thus, A and C have same elements but some elements of A and B are different.
So, the sets A and C are equal sets but sets A and B are not equal.
Two sets A and C are said to be equal if every element in A belongs to C and every element in C belongs to A.
If A and C are equal sets, than we write A = C.
*****************************************
Example-6. Consider the following sets :
A = {p,q,r} B={q,p,r}
sol) In the above sets, every element of A is also an element of B
.^. A ⊂ B
Similarly every element of B is also in A.
.^. B⊂ A
Thus, we can also write that if
B⊂ A and A⊂ B <=> A = B.
Here <=> is the symbol for two way implication and is usually read as, if and only if (briefly written as "iff").
*****************************************
Examples-7. If A={1,2,3,....} and N is a set of natural numbers, check whether A and N are equal?
sol) The elements are same in both the sets. Therefore, both A and N are the set of Natural numbers.
Therefore the sets A and N are equal sets or A=N.
*****************************************
Example-8. Consider the sets A ={p,q,r,s} and B={1,2,3,4}. Are they equal?
sol) A and B do not contain the same elements.
So, A=/= B.
*****************************************
Example-9. Let A be the set of prime numbers less than 6 and P the set of prime factors of 30. Check if A and P are equal.
sol) The set of prime numbers less than 6
A={2,3,5}
The prime factors of 30 are 2,3 and 5. So,
P={2,3,5}
Since the elements of A are the same as the elements of P, therefore, A and P are equal.
*****************************************
Example-10. Show that the sets A and B are equal, where
A = {x:x is a letter in the word 'ASSASSINATION'}
B = {x : x is a letter in the word STATION}
sol) Given,
A={x:x is a letter in the word ' ASSASSINATION'}
This set A can also be written as A={A,S,I,N,T,O} since generally elements in a set are not repeated.
Also given that
B = { x : x is a letter in the word STATION}
'B' can also be written as B = { A,S,I,N,T,O}
So, the elements of A and B are same and
A = B
*****************************************
Exercise - 2.3
1) Which of the following sets are equal?
(1) A = { x: x is a letter in the world FOLLOW}
(2) B = { x: x is a letter in the FLOW }
(3) C = { x : x is a letter in the word WOLF }
Sol ) All the above sets are in set-builder form.Lets convert into roster form
Te roster form of set A = { F , O , L , W }
(since same elements in a set cannot be repeated)
B Roster form of Set B = { F, L, O W }
C= Roster form of Set C = { W, O,L.F}
All the three sets A, B and C have same elements { F,L,O,W}.
So the three sets are equal. A=B=C
---------------------------------------------------------------------------
2.) Consider the following sets and fill up the blanks in the statement given below with "=" or
=/= as to make the statement true.
A) = {1,2,3}
B = { the first three natural numbers }
C= { a,b,c,d}
D = { d,c,a,b}
E = { a,e,i,o,u}
F = { set of vowels in English Alphabet }
(1) A ---- B
Sol) { 1,2,3 } ----- { the first three natural numbers }
=> { 1,2,3} ---- { 1,2,3}
=> Set A = Set B
=> A = B
-----------------------------------------------------------------------------
(2) A ---- E
sol) { 1,2,3}------ { a,e,i,o,u}
=> { natural no} ---- { alphabet}
=> Set A =/= Set E
=> A =/= B
------------------------------------------------------------------------------------
(3) C ---- D
Sol) { a,b,c,d} ----- { d,c,a,b}
=> Set C = Set D
=> C = D
--------------------------------------------------------------------------------
(4) D ------ F
Sol) {d,c,a,b} ----- { set of vowels in English Alphabet }
=> {d,c,a,b} ----- { a,e,i,o,u}
Both sets are different
=> Set D =/= Set F
=> D =/= F
--------------------------------------------------------------------------------
(5) F ---- A
Sol ) { set of vowels in English Alphabet } ------ { 1,2,3}
=> {a,e,i,o,u } ----- {1,2,3}
=> Set F =/= Set A
=> F =/= A
-------------------------------------------------------------------------------
(6) D --- E
Sol) { d,c,a,b } ---- { a,e,i,o,u}
=> Set D =/= Set E
=> D =/= E
-------------------------------------------------------------------------------
(7) F ---- B
Sol) { set of vowels in English Alphabet } ---- { The first three natural numbers }
=> { a,e,i,o,u } ---- { 1,2,3}
=> Set F =/= Set B
=> F =/= B
*****************************************************
3) In each of the following, state whether A= B or not.
(1) A = { a,b,c,d } B = { d,c,a,b}
Sol) A = B
Since both the Sets A and B have same elements {a,b,c,d}
----------------------------------------------------------------------------
(2) A = { 4,8,12,16} B = { 8,4,16,18}
Sol) A =/= B
Since not all elements are same in both sets A and B except { 4,8,16}
===================================================
(3) A = { 2,4,6,8,10} B= { x : x is a positive even integer and x<10}
Sol ) A =/= B
Since B { even integer but x < 10 } = { 2,4,6,8,}
All elements are same except {10} in set A
-------------------------------------------------------------------------------------
4) A = { x : x is a multiple of 10 } B = { 10,15,20,25,30}
Sol ) A =/= B
set A { 10,20,30,40.......} B = { 10,15,20,25,30}
Both Sets are different
********************************************************
4) State the reasons for the following.
(1) { 1,2,3......10} =/= { x : x belongs N and 1< x < 10 }
Sol ) Check Set B {x : x belongs N and 1<x<10}
"x belongs to N ( natural no) :- {1,2,3,4....}
" 1<x<10 " :- "x is greater than 1 x>1 but less then "10" x<10
Set A { 1,2,3...10 } =/= Set B { 2(x> 1), 3, 4..... 9 ( x < 10) }
reason :- Since Set A is a set of natural no's starting from "1" and ends at "10" whereas Set B also a set of natural no's but it starts with "2" and ends at "9" ...so both sets are different.
.^. {1,2,3,.....10} =/= { 2,3,4,,,,,9}
---------------------------------------------------------------------------------
(2) { 2,4,6,8,10 } =/= { x : x = 2n+1 and x belongs N}
Sol ) Set B = { x : x = 2n +1 and x belongs N}
2n+1 :-
2(0) +1 = 0+1 = 1
2(1) +1 = 2+1 = 3
2(2) + 1 = 4 +1 = 5
2( 3 ) + 1 = 6+1 = 7
2 ( 4 ) + 1 = 8 +1 = 9
" x= 2n+1" :- ( 1,3,5,7,9....)
Set A { 2,4,6,8,10} are even natural numbers
Set B { 1,3,5,7,.....) are set of odd natural numbers
Set A { 2,4,6,8,10} =/= Set B { 1,3,5,7,9....}
Reason :- Set A is a set of even natural no's whereas Set B is on odd natural no's..both sets are different.
.^. { 2,4,6,8,10} =/= { 1,3,5,7,9....]
-------------------------------------------------------------------------------------
3) { 5,15,30,45 } =/= { x:x is a multiple of 15}
Sol) Set B { x : x is a multiple of 15} = { 15 ,30,45 ,60.....}
Set A { 5,15,30,45 } =/= Set B { 15,30,45,60....}
Reason :- Since {5} does not exists in Set B
.^. {5,15,30,45} =/= { 15,30,45,60}
---------------------------------------------------------------------------------
4) {2,3,5,7,9} =/= { x:x is a prime number }
Sol ) Set B { x ; x is a prime number } = { 2,3,5,7,11...}
Set A { 2,3,5,7,9} =/= Set B { 2,3,5,7,11...]
Reason :- Since 9 is not a prime number which does not exists in set b
.^. {2,3,5,7,9} =/= {2,3,5,7,11...}
*********************************************************
5.) List all the subsets of the following sets.
(1) B = {p,q}
Sol) Subsets with one element { p }, { q}
Subsets with two elements { p,q}
Empty set is also a subset {
Thus the list of all subsets of {p, q} will be.
{p} , {q} , {
p,q}, {
Note :- if you want to check whether you found all subsets or not just use
2^ n formula where n= no of elements
example { p , q} = 2 elements
2^2 = 4
------------------------------------------------------------------------------------------
(2) C = { x,y,z }
Sol ) Subsets with one element { x}, { y }, { z }
Subsets of two elements { x,y}, { x, z }, { y, z }
Subsets of two three elements { x,y,z}
Empty set {
Thus, the lists of { x,y,z } all subsets will be
{ x } , { y }, { z },
{ x,y },{x,z}, {y,z }, {
x, y, z}, {
2^3 = 2*2*2=8 elements
-------------------------------------------------------------------------------------
(3) D = { a,b,c,d}
Sol ) Subsets with one element { a }, { b }, { c }, { d }
Subsets with two elements { a,b }, { a,c } , { a,d }, { b , c } , {b,d } , { c, d }
Subsets with three elements { a,b,c } , {a,b,d }, {a,c,d }, {b,c,d }
Subsets with four elements { a,b,c,d }
Null or empty set {
Thus, the lists of { a,b,c,d } all subsets will be.
{ a }, { b } , { c }, {d },
{ a, b }, { a ,c } , { a,d }, b, c } , { b, d } , { c , d},
{ a,b,c }, { a, b,d }, { a, c d }, { b,c,d }, {
a,b,c,d }, {
2^ 4 = 2*2*2*2 = 16 elements
--------------------------------------------------------------------------------
4) E = { 1,4,9,16 }
Sol ) One elements subset { 1 }, { 4 }, { 9 }, { 16 }
Two element subsets { 1,4 } , { 1,9 } , { 1, 16 } , { 4, 9 }, { 4, 16 }, { 9, 16 }
3 elements subsets { 1,4,9 }, { 1,4,16 }, {1,9, 16 }, { 4,9, 16}.
4 elements subsets { 1,4,9,16 }
Null or empty set {
Thus the lists of {1,4,9,16} all subsets will be.
{ 1 }, { 4 }. {9 }, { 16}
{ 1,4 }, { 1,9 }, { 1, 16 }, { 4,9 }, { 4,16 }, { 9,16}
{ 1,4,9 }, { 1,4,16 }, 1,9,16}, {4,9,16}
{ 1,4,9, 16 } , {
------------------------------------------------------------------------------------
5) F = { 10, 100, 1000 }
Sol ) 1 element subset { 10 }, { 100 }, { 1000 }
2 element subsets { 10, 100 }, { 10, 1000 }, { 100, 1000 }
3 elements subsets { 10, 100, 1000 }
Null or empty set {
List of { 10, 100, 1000 } all subsets is
{ 10 }, { 100 }, { 1000}
{ 10, 100 }, { 10, 1000}, { 100, 1000}
{ 10, 100, 1000}, { } = Total 8 subsets
************************************************************
Consider the set E={2,4,6} and F ={6,2,4}
Note that E=F. Now, since each element of E also belongs to F, therefore E is a subset of F.
But each element of F is also an element of E.
So F is a subset of E.
In this manner it can be shown that every set is a subset of itself.
if A and B contain the same elements, they are equal
i.e., A = B.
By this observation we can say that
"Every set is subset of itself"
*****************************************
Example-11. Consider the sets ∅,
A={1,3}
B = {1,5,9}
C = {1,3,5,7,9}.
Insert the symbol ⊂ or ⊄ between each of the following pair of sets.
(!) ∅.......B
sol) ∅ ⊂ B, as ∅ is a subset of every set.
******************************************
(!!) A......B
sol) A ⊄B, for 3∈ A but 3 ∉ B.
*****************************************
(!!!) A..........C
sol) A⊂ C as 1,3 ∈ A also belong to C.
*****************************************
(!v) B......C
sol) B ⊂ C as each element of B is also an element of C.
******************************************
consider the following sets
A={1,2,3}
B={1,2,3,4}
C={1,2,3,4,5}
All the elements of A are in B
.^. A⊂B.
All the elements of B are in C
.^. B⊂C
All the elements of A are in C
.^. A⊂C
That is ,
A⊂B,
B⊂C => A⊂ C
*****************************************
Exercise-2.4
1). State which of the following statements are true given that,
A={1,2,3,4}
(!) 2∈A
sol) True
******************************************
(!!) 2 ∈ {1,2,3,4}
sol) True
*****************************************
(!!!) A ⊂ {1,2,3,4}
sol) True
******************************************
(!v) {2,3,4} ⊂ {1,2,3,4}
sol) True
*****************************************
(*) State the reasons for the following
(!) {1,2,3....10} =/= {x:x∈N and 1<x<10}
sol) R.H.S condition:
x ∈ N = (1,2,3,4.....)
1<x<10 = (2.....9)
(*) L.H.S : {1,2,3....10}
Since L.H.S does not satisfy the condition on R.H.S as 1 and 10 are excluded.
*****************************************
(!!) {2,4,6,8,10} =/= {x:x=2n+1 and x∈N}
sol) R.H.S condition :
(*) 1st cond:-x = 2n+1
(2(1)+1=3)
(2(2)+1=5)
(2(3)+1=7)
x= 2n+1 => (3,5,7,......)
2nd cond:-x ∈ N => (1,2,3,4...)
(*) L.H.S Condition:
{2,4,6,8,10}
Now,
In the above L.H.S set should be {3,5,7,9...} instead of {2,4,6,8,10}
.^. L.H.S =/= R.H.S
*****************************************
(!!!) {5,15,30,45} =/= {x:x is a multiple of 15}
sol) R.H.S : x is a multiple of 15
ex : 1*15=15, 2*15=30, 3*15=45 so on..
But Here
L.H.S: {5,15,30,45}...5 is not multiple of 15
L.H.S =/= R.H.S
*****************************************
(!V) {2,3,5,7,9} =/= { x:x is a prime number}
sol) Prime number : which divisible by 1 and itself
example:- ( 2,3,5,7,11,13,15.......)
Now,
L.H.S : {2,3,5,7,9.....} here 9 is not prime.
Hence
L.H.S =/= R.H.S
*****************************************
3. List all the subsets of the following sets.
(!) B = {p,q}
sol) No.of elements in set n = 2
No of subsets = 2^n = 2^2 = 4
{ },{p} {q}, {p,q} = 4
Note: Null set { } is the subset of set itself.
******************************************
(!!) C = {x,y,z}
sol) No.of elements n= 3
No. of subsets = 2^3 = 8
Now
{ }, {x},{y},{z},{x,y},{x,z},{y,z},{x,y,z} = 8
******************************************
(!!!) D = {a,b,c,d}
sol) No.of elements n= 4
No.of subsets = 2^4 = 16
{ }, {a}, {b}, {c}, {d}
{a,b}, {a,c}, {a,d},
{b,c}, {b,d},
{c,d}
{a,b,c},{a,b,d},{a,c,d},{b,c,d}
{a,b,c,d}
************
subsets= 16
************
*****************************************
(!v) E = {1,4,9,16}
sol) No. of elements n= 4
No. of subsets = 2^4 = 16
subsets:-
{ }, {1 }, {4}, {9}, {16}
{1,4}, {1,9}, {1,16},{4,9},{4,16},{9,16}
{1,4,9}, {1,4,16}, {1,9,16},{4,9,16}
{1,4,9,16}
******************
Total subsets = 16
*****************
******************************************
(V) F = { 10,100,1000}
sol) No. of elements n = 3
No. Of subsets = 2^3 = 8
Subsets:
{ }, {10}, {100},{1000}
{10,100},{10,1000}, {100,1000}
{10,100,1000}
**************
total subsets = 8
***************
*****************************************
1) State which of the following sets are empty and which are not?
(1) The set of straight lines passing through a point.
Sol) Not empty , Because if a line pass through any point (x,y)
----------------------------------------------------------------------------------
(2) Set of odd natural numbers divisible by 2.
Sol) Empty .
Natural numbers : 1,2,3,4,5,6,7,8,9,10,11.....so.on
odd natural numbers :- 1,3,5,7,9,11.... is any of this odd.no's divisible by "2"?
An odd number is an integer that is not evenly divisible by '2'.
Therefore its is an empty set
----------------------------------------------------------------------------------------
(3) {x:x is a natural number, x<5 and x>7}
Sol) Empty.
natural number x<5 : 1,2,3,4.
natural number x>7 : 8,9,10,11
Is there a number which is less than 5 and greater than 7 simultaneously?NO
Therefore its an empty set.
---------------------------------------------------------------------------------------
(4) {x:x is a common point to any two parallel lines.}
Sol) Empty
Parallel Lines :- Two lines is said to be parallel if an only if they have no common points.
<---------------------------->
<---------------------------->
(parallel lines)
Therefore it is an empty set.
----------------------------------------------------------------------------------------
5) Set of even prime numbers.
Sol) Not empty.
Prime Numbers :- 2,3,5,7,11,........
Even Prime Numbers :- A prime number which is divisible by "2"
We have "2" which is even as well as prime number.
.^. It is not an empty set
========================================================
Finite Set :- if There are only limited number of elements present in a set then such a set is called finite set.
Example :- L= {p,q,r,s} It has only "4" element which is finite
(*) Infinite Set :- if there is an infinite number of elements present in a set. then such a set is called an infinite set.
Example :- A = {x:x is an even number} .
Set A have all even numbers .We cannot count all of them
2. Which of the following sets are finite and infinite.
(1) The sets of months in a year.
Sol) There are only 12 months in a year.. This set has only 12 elements.
.^. the set is finite.
(2) {1,2,3,.....99,100 }
sol) Clearly finite.
(3) The set of prime numbers less than 99.
Sol) Finite.
----------------------------------------------------------------------------------
3) State whether each of the following set is finite or infinite.
(1) The set of letters in the English alphabet.
Sol) Finite. (A,B,C..........Z)
(2) The set of lines which are parallel to the X-axis.
Sol) Infinite.
(3) The set of numbers which are multiples of 5.
Sol) Infinite .
.( 5*1=5...5*2=10 ....5*3=15..........so on)
(4) The set of circles passing through the origin (0,0)
Sol) Infinite. ( infinite number of circles can pass through origin).
*************************************************************
(*) Venn Diagrams
Venn-Euler diagram or simply Venn-diagram is a way of representing relationship between sets. These diagrams consists of rectangles and closed curves usually circles.
The universal set is usually represented by a rectangle.
(!) Consider that μ ={ 1,2,3,.....10} is the universal set of which A = { 2,4,6,8,10} is a subset. Then the venn-diagram is as:
(!!) μ = { 1,2,3,.....10} is the universal set of which , A ={2,4,6,8,10} and B={4,6} are subsets and also B ⊂ A. Then we have the following figure.
(!!!) Let A ={a,b,c,d} and B={c,d,e,f}. Then we illustrate these sets with a Venn diagram as.
(*) Basic operations on sets
We know that arithmetic has operations of additions, subtraction and multiplication of numbers.
Similarly in sets, we define the operation of union, intersection and difference of sets
(*) Union of sets
Example-12. Suppose A is the set of students in your class who were absent on Tuesday and B the set of students who were absent on Wednesday. Then,
A={Roja, Ramu, Ravi} and
B = {Ramu, Preethi, Haneef}
Now, we want to find K, the set of students who were absent on either Tuesday or Wednesday.
Then, does
Roja ∈ K?
Ramu ∈ K?
Ravi ∈ K?
Haneef ∈ K?
Preeti ∈ K?
Akhila ∈ K?
Roja, Ramu, Ravi, Haneef and Preeti all belong to K but Ganpati does not.
Hence,
K = {Roja, Ramu, Raheem, Prudhvi,Preethi}
Here K is called the union of sets A and B. The union of A and B is the set which consists of all the elements of A and B and the common elements being taken only once. The symbol ⋃ is used to denote the union.
Symbolically, we write A ⋃ B and usually read as 'A union B'
A ⋃ B = { x : x∈A or x∈B }
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Example-13. let A= {2,5,6,8} and B={5,7,9,1}. Find A⋃ B
sol) A ⋃ B = { 1,2,5,6,7,8,9}
Note: common element 5 was taken only once while writing A ⋃ B
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Example-14. let A= {a,e,i,o,u} and B={a,i,u}. Show that A ⋃ B = A.
This example illustrate that union of sets A and its subset B is the Set A itself.
i.e, if B ⊂A, then A⋃ B = A.
The union of the sets can be represented by a Venn-diagram as shown(shaded portion)
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Example-15. Illustrate A⋃ B in Venn-diagram where.
A= {1,2,3,4} and B = {2,4,6,8}
sol) A ⋃ B = { 1,2,3,4,6,8}
(*) intersection of sets.
Let us again consider the example of absent students. This time we want to find the set L of students who were absent on both Tuesday and Wednesday.
We find that L = { Ramu}.
Here, L is called the intersection of sets A and B.
In general, the intersection of sets A and B is the set of all elements which are common to A and B.i.e., those elements which belongs to A and also belong to B.
We denote intersection by A∩B. ( read as "A intersection B"). Symbolically , we write.
A∩B = { x : x∈A and x∈B}
The intersection of A and B can illustrated in the Venn-diagram as shown in the shaded portion in the adjacent figure.
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Example-16. Find A∩ B when A={5,6,7,8} and B={3,4,5}
sol) The common elements in both A and B are 7,8.
.^. A ∩ B = { 7,8}
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Example-17. Illustrate A∩B in Venn-diagram where A={1,2,3} and B = {3,4,5}
sol)
.^. A ∩ B = {3}
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(*) Disjoint Set
Suppose A={1,3,5,7} and B={2,4,6,8} .
We see that there are no common elements in A and B. Such sets are known as disjoint sets.
The disjoint sets can be represented by means of the Venn-diagram as follows.
A ∩ B = ∅
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(*) Difference of Sets
The difference of sets A and B is the set of elements which belong A but do not belong to B.
We denote the difference of A and B by A-B or simply " A minus B".
A- B = { x : x ∈ A and x ∈ B}
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Example-18. Let
A = { 1,2,3,4,5,6},
B= { 4,5,6,7}
Find A - B
sol) Given A ={ 1,2,3,4,5} and B={4,5,6,7} . Only the elements which are in A but not in B should be taken.
A - B = {1,2,3}
Since 4,5 are the elements in B are not taken.
Similarly for B-A, the elements which are only in B are taken.
.^. B - A = { 6,7}
(4,5 are the elements in A)
Note that A - B =/= B - A
The Venn-diagram of A - B is as shown
Example-19. Observe the following
A = {3,4,5,6,7}
.^. n(A) = 5
B = {1,6,7,8,9}
.^. n(B) = 5
A⋃ B = { 1,3,4,5,6,7,8,9}
.^. n(A⋃ B) = 8
A ⋃B = { 6,7}
.^. n(A⋃B) = 2
.^. n(A ⋃ B) = 5 + 5 - 2 = 8
We observe that
n(A⋃ B) = n(A) + n(B) - n (A B)
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Exercise- 2.5
1). If A={1,2,3,4}; B={1,2,3,5,6} then find A∩B and B∩A . Are they equal?
sol) A∩B = { 1,2,3,4}∩ { 1,2,3,5,6}
common terms in both sets
.^. A ∩ B = { 1,2,3}
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B ∩ A = {1,2,3,5,6} ∩ {1,2,3,4}
B ∩ A = {1,2,3}
A ∩ B and B ∩ A are equal.
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2). A={0,2,4}, find A ∩∅ and A∩A. Comment.
sol). A ∩∅={0,2,4} ∩ { }
=> A ∩ ∅ = { }
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A∩A = { 0,2,4} ∩ {0,2,4}
=> {0,2,4}
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3). If A={2,4,6,8,10} and B={3,6,9,12,15},
find A-B and B-A.
sol)A - B = {2,4,6,8,10} - {3,6,9,12,15}
=> {2,4,8,10}
B-A={3,6,9,12,15} - {2,4,6,8,10}
=> {3,9,12,15}
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4). If A and B are two sets such that A⊂B then what is A ∪B?
sol) Let
A = {1,2,3,4}
B = {1,2,3,4,5,6}
A⊂ B (Every element of A is in B)
Now,
A ⋃ B = {1,2,3,4} ⋃ {1,2,3,4,5,6}
=> { 1,2,3,4,5,6} = B
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5). If A={x : x is a natural number }
B={x:x is an even natural number}
C={x:x is an odd natural number}
D={x :x is a prime number}
find
first, let's write all the sets in roster form.
A={x:x is natural number}
= {1,2,3,4,5,6,7...}
B={x : x is an even natural number}
= {2,4,6,8,10,12,14.....}
C = { x : x is an odd natural number}
= { 1,3,5,7,9....}
D = {x : x is prime number}
= { 2,3,5,7,11,13,15,17.....}
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(!) A∩B
sol) Common elements in A & B
{natural numbers} ∩ {even natural numbers}
{1,2,3,4,5,6,7...} ∩ {2,4,6,8,10,12,..}
{2,4,6,8,.......} = B (even natural number)
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(!!) A∩C
sol) Common elements between A & C
{natural number} ∩ { odd natural number}
{ 1,2,3,4,5,6,7...} ∩ {1,3,5,7,9,11,13,..}
{1,3,5,7,...} = C (odd natural number)
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(!!!) A∩D
sol) common elements in A & D
{natural number} ∩{prime number}
{1,2,3,4,5,6,7...} ∩ {2,3,5,7,11,13,15,17..}
{2,3,5,7....} = D (prime number)
*************************************** (!V) B∩C,
sol) Common elements in B&C
{even natural number}∩{odd natural number}
{ 2,4,6,8,10,12,14...} ∩ { 1,3,5,7,9...}
{ } = ∅ (empty set)
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(V) B∩D
sol) {even natural number}∩{prime number}
{2,4,6,8,10....} ∩ {2,3,5,7,11...}
{ 2 }
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(V!) C∩D.
sol) {odd natural number} ∩ { prime number}
{ 1,3,5,7,9,11,13...} ∩ { 2,3,5,7,11,13...}
{ 3,5,7,11,13....}
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6). If
A={3,6,9,12,15,18,21}
B={4,8,12,16,20}
C= {2,4,6,8,10,12,14,16}
D={5,10,15,20}
find
(!) A - B
sol) {3,6,9,12,15,18,21}- {4,8,12,16,20}
= { 3,6,9,15,18,21}
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(!!) A - C
sol) {3,6,9,12,15,18,21} - {2,4,6,8,10,12,14,16}
A - C = {3,9,15,18,21}
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(!!!) A - D
sol) {3,6,9,12,15,18,21}-{5,10,15,20}
A - D = {3,6,9,12,18,21}
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(!v) B - A
sol) {4,8,12,16,20} - {3,6,9,12,15,18,21}
B - A = { 4,8,16,20}
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(V) C - A
sol) { 2,4,6,8,10,12,14,16} - {3,6,9,12,15,18,21}
C - A ={2,4,8,10,14,16}
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(V!) D - A
sol) {5,10,15,20} - {3,6,9,12,15,18,21}
D - A = {5,10,20}
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(V!!) B- C
sol) {4,8,12,16,20} - {2,4,6,8,10,12,14,16}
B - C = {20}
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(V!!!) B-D
sol) {4,8,12,16,20} - { 5,10,15,20}
B - D = {4,8,12,16}
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(!x) C - B
sol) {2,4,6,8,10,12,14,16} - { 4,8,12,16,20}
C - B = {2,6,10,14}
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(x) D - B
sol) {5,10,15,20} - { 4,8,12,16,20}
D - B = {5,10,15}
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7. State whether each of the following statement is true or false. Justify your answers.
(!) { 2,3,4,5} and {3, 6} are disjoint sets.
sol) Two sets are disjoint if they have no common element.
We know :- intersection of two sets means common element in both sets
Now,
{2,3,4,5} ∩ {3,6}
= { 3} ≠ ∅
Since, there is a common element in both set.
The given sets are not disjoint.
So, the given statement is False.
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(!!) {a,e,i,o,u} and {a,b,c,d} are disjoint sets.
sol) {a,e,i,o,u} ∩ {a,b,c,d}
{a} ≠ ∅
Since, there is a common element in both set.
The given sets are not disjoint.
So, the given statement is False.
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(!!!) {2,6,10,14} and {3,7,11,15} are disjoint sets.
sol) No common elements in both sets
Hence ,its disjoint
So, the given statement is True
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(!V) { 2,6,10} and {3,7,11} are disjoint sets.
sol) no common elements in both sets
Hence pair is disjoint
So, the given statement is True.
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