# 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.

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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