Question

Determine whether the following relation is reflexive, symmetric and transitive:

Relation R in the set Z of all integers defined as R = {(x, y) : x − y is an integer}.

Answer 1

R = {(x, y) : x − y is an integer}

Now, for every x ∈ Z, (x, x) ∈ R as x − x = 0 is an integer.

∴ R is reflexive.

Now, for every x, y ∈ Z, if (x, y) ∈ R, then x − y is an integer.

⇒ −(x − y) is also an integer.

⇒ (y − x) is an integer.

∴ (y, x) ∈ R

∴ R is symmetric.

Now, let (x, y) and (y, z) ∈ R, where x, y, z ∈ Z. 

⇒ (x − y) and (y − z) are integers.

⇒ x − z = (x − y) + (y − z) is an integer.

∴ (x, z) ∈ R

∴ R is transitive.

Hence, R is reflexive, symmetric and transitive.