Q. 54.8( 4 Votes )

# Let Z be the set of integers. Show that the relation R = {(a, b) : a, b ∈ Z and a + b is even} is an equivalence relation on Z.

Answer :

We have,

Z = set of integers and

R = {(a, b) : a, b ∈ Z and a + b is even} be a relation on Z.

To prove: 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 is even

⇒ (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 even

⇒ b + a is even

⇒ (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) ∈ R and (b, c) ∈ R For some a, b, c ∈ Z

⇒ a + b is even and b + c is even

[if b is odd, then a and c must be odd ⇒ a + c is even,

If b is even, then a and c must be even ⇒ a + c is even]

⇒ a + c is even

⇒ (a, c) ∈ R

⇒ R is transitive

Hence, R is an equivalence relation on Z

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