Let A = { 1 , 2 , 3 } , we define (i) R 1 = { ( 1 , 1 ) , ( 2 , 2 ) , ( 3 , 3 ) } (ii) R 2 = { ( 1 , 2 ) , ( 2 , 1 ) , ( 2 , 2 ) , ( 1 , 1 ) } (iii) R 3 = { ( 1 , 2 ) , ( 1 , 3 ) } (iv) R 4 = { ( 1 , 1 ) , ( 3 , 3 ) } (v) R 5 = A × A then, which of the following is true?

The statement R 1 and R 5 both are equivalence relation

R 1 , R 2 , R 3 , R 5 are all equivalence relations

Let A = { 1 , 2 , 3 } , we define (i) R 1 = { ( 1 , 1 ) , ( 2 , 2 ) , ( 3 , 3 ) } (ii) R 2 = { ( 1 , 2 ) , ( 2 , 1 ) , ( 2 , 2 ) , ( 1 , 1 ) } (iii) R 3 = { ( 1 , 2 ) , ( 1 , 3 ) } (iv) R 4 = { ( 1 , 1 ) , ( 3 , 3 ) } (v) R 5 = A × A then, which of the following is true?

The statement R 1 and R 5 both are equivalence relation

Check whether the relation R defined in the set 1 , 2 , 3 , 4 , 5 , 6 as R = a , b : b = a + 1 is reflexive, symmetric or transitive.

Suppose A = 1 , 2 , 3 , 4 , 5 , 6 . A relation R is defined on set A as: R = { ( a , b ) : b = a + 1 } ∴ R = { ( 1, 2 ) , 2 , 3 , 3 , 4 , 4 , 5 , 5 , 6 } Reflexive Property : A relation R in a set A is said to be reflexive if x , x ∈ R , for every x ∈ A Now, 1 , 1 , 2 , 2 , 3 , 3 , 4 , 4 , 5 , 5 , 6 , 6 ∉ R ∴ R is not reflexive. Symmetric Property : A relation R in a set A is said to be symmetric if x , y ∈ R implies that y , x ∈ R for all x , y ∈ A . Now it can be observed that 1 , 2 ∈ R , but 2 , 1 ∉ R . Hence , R is not symmetric. Transitive Property : A relation R in a set A is said to be transitive if x , y ∈ R and y , z ∈ R implies that x , z ∈ R for all x , y , z ∈ A . Now, 1 , 2 , 2 , 3 ∈ R but, ( 1 , 3 ) ∉ R ∴ R is not transitive. Hence, R is neither reflexive, nor symmetric, nor transitive.