JEE Mathematics/ Lakshmikanta Satapathy/ Relations and Functions theory part 2/ Types of relations/ Reflexive Symmetric and Transitive relations/ Equivalence relation/ Equivalence class

1. Physics Helpline
L K Satapathy
Relations & Functions Theory 2

2. Physics Helpline
L K Satapathy
Types of Relations :
Relations & Functions Theory 2
1. Empty Relation : A relation R in a set A is an Empty Relation , if no element
of A is related to any element of A  R =   AA
Illustration :
Consider the set A of all students of a Girls School
Consider the relation R = { (x , y) : y is a brother of x ; x , y  A }
We observe that , no student of the given School is a Male
 No pair of students can satisfy the Relation
 No student can be the brother of any other student
 R is an empty Relation

3. Physics Helpline
L K Satapathy
Relations & Functions Theory 2
2. Universal Relation : A relation R in a set A is an Universal Relation , if each
element of A is related to every element of A  R = A  A
Illustration :
Consider the set A = { 1 , 3 , 5 , 7 , 9 , 11 }
Consider the relation R = { (x , y) : x + y is an even number ; x , y  A }
We observe that , every element of set A is an odd number
 Each element of A is related to every element of A
And the sum of two odd numbers is an even number
 R is an Universal Relation

4. Physics Helpline
L K Satapathy
Relations & Functions Theory 2
3. Reflexive Relation : A relation R in a set A is Reflexive if
( , )x x R for all x A 
Illustration :
Consider the set A = { 1 , 2 , 3 , 4 }
Consider the relation R = {(1 , 1) , (1 , 2) , (2 , 2) , (2 , 3) , (3 , 3) , (4 , 4)}
We observe that each of (1 , 1) , (2 , 2) , (3 , 3) and (4 , 4)  R
 Each element of A is related to itself
 R is a Reflexive Relation
 (x , x)  R for all x  A

5. Physics Helpline
L K Satapathy
Relations & Functions Theory 2
4. Symmetric Relation : A relation R in a set A is Symmetric if
( , ) ( , ) ,x y R y x R for all x y A   
Illustration :
Consider the set A = { 1 , 2 , 3 , 4 }
Consider the relation R = {(1 , 1) , (1 , 2) , (2 , 2) , (2 , 3) , (2 , 1) , (3 , 2)}
We observe that (1 , 2)  R  (2 , 1)  R and (2 , 3)  R  (3 , 2)  R
 For each (x , y)  R , we have (y , x)  R
 R is a Symmetric Relation
It may be noted that R is not Reflexive as 3  A but (3 , 3)  A
[ One example is enough to falsify a statement ]

6. Physics Helpline
L K Satapathy
Relations & Functions Theory 2
5. Transitive Relation : A relation R in a set A is Transitive if
( , ) ( , ) ( , ) , ,x y R and y z R x z R for all x y z A    
Illustration :
Consider the set N of natural numbers
Consider the relation R = { (x , y) : x < y ; x , y  N }
 No Natural number is less than itself  R is NOT Reflexive
 If x is less than y , then y is NOT less then x  R is NOT Symmetric
 If x  y and y  z , then x  z  (x , y)  R and (y , z)  R  (x , z)  R
 R is a Transitive Relation

7. Physics Helpline
L K Satapathy
Relations & Functions Theory 2
6. Equivalence Relation : A relation R in a set A is an Equivalence Relation
if it is Reflexive , Symmetric and Transitive
Illustration :
Consider the relation R = { (x , y) :  x – y  is divisible by 3 ; x , y  Z }
 For all x  Z ,  x – x  = 0 , which is divisible by 3  R is Reflexive
 If  x – y  is divisible by 3 , then  y – x  is divisible by 3  R is Symmetric
If  x – y  is divisible by 3 and  y – z  is divisible by 3 ,
then  x – z  is divisible by 3  R is Transitive
 R is an Equivalence Relation
Consider the set Z of Integers

8. Physics Helpline
L K Satapathy
Relations & Functions Theory 2
Equivalence Class :
If a Relation R in a Set A , which divides A into mutually disjoint Subsets Ai ,
called Partitions , satisfying the following conditions :
(i) All elements of Ai are related to each other for all i
(ii) No element of Ai is related to any element of Aj , i  j
(iii)  Ai = A and Ai  Aj =  , i  j
Then the Subsets Ai are called equivalent Classes

9. Physics Helpline
L K Satapathy
Relations & Functions Theory 2
We observe that
(i) (x – 0) is divisible by 2 , when x = 0 ,  2 ,  4 ,  6 , . . .
 We say that all even integers are related to (0) through R
 The equivalence class related to (0) is given by
A1 = [ 0 ] = { . . . – 6 , – 4 , – 2 , 0 , 2 , 4 , 6 . . . }
(ii) (x – 1) is divisible by 2 , when x =  1 ,  3 ,  5 , . . .
 We say that all odd integers are related to (1) through R
 The equivalence class related to (1) is given by
A2 = [ 1 ] = { . . . – 5 , – 3 , – 1 , 1 , 3 , 5 , . . . }
Example : Consider a Relation R in the Set Z of integers , given by
R = { (x , y) : x – y is divisible by 2 ; x , y  Z }

10. Physics Helpline
L K Satapathy
Relations & Functions Theory 2
Testing conditions :
A1 = { . . . – 6 , – 4 , – 2 , 0 , 2 , 4 , 6 . . . } (set of EVEN integers)
A2 = { . . . – 5 , – 3 , – 1 , 1 , 3 , 5 , . . . } (set of ODD integers)
(iii) A1  A2 = Z and A1  A2 = 
(i) All elements of A1 are related to each other
and all elements of A2 are related to each other
(ii) No element of A1 is related to any element of A2
and no element of A2 is related to any element of A1
 A1 and A2 are the Equivalence Classes of Z partitioned by R

11. Physics Helpline
L K Satapathy
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