Question

Check if R: Z → Z, R = {(a, b) |2 divides
a-b} is equivalence relation.​

Answer 1

SOLUTION

TO CHECK

Check if R: Z → Z, R = {(a, b) |2 divides a-b} is equivalence relation.

EVALUATION

Here the given relation is

R : Z → Z such that R = {(a, b) |2 divides a -b}

CHECKING FOR REFLEXIVE

Let a ∈ Z

Since 2 divides a - a

So (a, a) ∈ R

So R is Reflexive

CHECKING FOR SYMMETRIC

Let a, b ∈ Z and (a, b) ∈ R

⇒2 divides a - b

⇒2 divides - ( b - a )

⇒2 divides ( b - a )

⇒(b, a) ∈ R

Thus (a, b) ∈ R implies (b, a) ∈ R

So R is symmetric

CHECKING FOR TRANSITIVE

Let a, b, c ∈ Z

Also let (a, b) ∈ R and (b, c) ∈ R

⇒2 divides a - b and 2 divides b - c

⇒2 divides ( a - b + b - c )

⇒2 divides ( a - c )

⇒(a, c) ∈ R

Thus (a, b) ∈ R and (b, c) ∈ R implies (a, c) ∈ R

R is transitive

Hence R is an equivalence relation

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