Q. 35.0( 1 Vote )

# Prove that the relation R on Z defined by(a, b) ∈ R ⇔ a – b is divisible by 5is an equivalence relation on Z.

Answer :

We have,

R = {(a, b) : (a – b) is divisible by 5} on Z.

We want to prove that R is an equivalence relation on Z.

Proof :

To prove that relation is equivalence, we need to prove that it is reflexive, symmetric and transitive.

Reflexivity : For Reflexivity, we need to prove that-

(a, a) R

Let a Z

a – a = 0

a – a is divisible by 5.

(a, a) R so R is reflexive

Symmetric : For Symmetric, we need to prove that-

If (a, b) R, then (b, a) R

Let (a, b) R

a – b = 5p(say) For some p Z

b – a = 5 × (–p)

b – a is divisible by 5

(b, a) R, so R is symmetric

Transitive : : For Transitivity, we need to prove that-

If (a, b) R and (b, c) R, then (a, c) R

Let (a, b) R and (b, c) R

a – b = 5p(say) and b – c = 5q(say), For some p, q Z

a – c = 5 (p + q)

a – c is divisible by 5.

R is transitive

R being reflexive, symmetric and transitive on Z.

R is equivalence relation on Z.

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