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Question

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.

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Solution

We observe the following properties of R.

Reflexivity:
Let a be an arbitrary element of Z. Then, aRClearly, a+a=2a is even for all aZ.a, aR for all aZSo, R is reflexive on Z.

Symmetry:
Let a, bRa+b is evenb+a is evenb, aR for all a, bZSo, R is symmetric on Z.

Transitivity:
Let a, b and b, cRa+b and b+c are evenNow, let a+b=2x for some xZand b+c=2y for some yZAdding the above two, we get a+2b+c=2x+2ya+c=2(x+y-b), which is even for all x, y, bZThus, a, cRSo, R is transitive on Z.

Hence, R is an equivalence relation on Z.

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