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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