Q. 15.0( 8 Votes )

Show that the rel

Answer :

We have,

R = {(a,b) : a–b is divisible by 3; a, b Z}

To prove : R is an equivalence relation

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 3

( 0 is divisible by 3).

(a, a) R

R is reflexive

Symmetric : For Symmetric, we need to prove that-

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

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

a – b is divisible by 3

a – b = 3p(say) For some p Z

–( a – b) = –3p

b – a = 3 × (–p)

b – a R

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, c Z and such that (a, b) R and (b, c) R

a – b = 3p(say) and b – c = 3q(say) For some p, q Z

a – c = 3 (p + q)

a – c = 3 (p + q)

(a, c) R

R is transitive

Since, R is reflexive, symmetric and transitive

R is an equivalence relation.

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