Question

Let N be the set of integers. A relation R on N is defined as R = {(x,y) | xy > 0, x,y, ∈ N}. Then, which one of the following is correct? 

(a) R is symmetric but not reflexive 

(b) R is reflexive but not symmetric 

(c) R is symmetric and reflexive but not transitive 

(d) R is an equivalence relation

Answer 1

Correct option is (d)

Explanation :

∵ R = {(x,y) | xy > 0, x, y ∈ N}

Reflexive 

∵ x, y ∈ N 

∴ x, x ∈ N ⇒ x² > 0 

∴ R is reflexive. 

Symmetric 

∵ x, y ∈ N 

and xy > 0 ⇒ yx > 0

∴ R is also symmetric

Transitive 

∵ x,y, z ∈ N ⇒ xy > 0, yz > 0 ⇒ xz > 0 

∴ R is also transitive. 

Thus, R is an equivalence relation