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Question

Prove that the relation R on Z defined by
(a, b) ∈ R ⇔ a − b is divisible by 5
is an equivalence relation on Z.

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Solution

We observe the following properties of relation R.

Reflexivity:
Let a be an arbitrary element of R. Then,a-a = 0 = 0 × 5a-a is divisible by 5a, aR for all aZSo, R is reflexive on Z.

Symmetry:
Let a, bRa-b is divisible by 5a-b = 5p for some pZb-a = 5 -p Here, -pZ [Since pZ]b-a is divisible by 5b, aR for all a, bZSo, R is symmetric on Z.

Transitivity:
Let a, b and b, cRa-b is divisible by 5a-b = 5p for some ZAlso, b-c is divisible by 5b-c = 5q for some ZAdding the above two, we geta-b+b-c = 5p+5qa-c = 5 (p+q)a-c is divisible by 5Here, p+qZa, cR for all a, cZSo, R is transitive on Z.

Hence, R is an equivalence relation on Z.

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