Show that the relation congruence modulo 2 on the set Z (set of integers) is an equivalence relation. Also find the equivalence class of 1.

Hello student,

Congruence modulo 2 is an equivalence relation:
1. Reflexive: for every integer x, x−x = 0 is indeed even,so x ≡ x
(mod 2).
2. Symmetric: if x ≡ y (mod 2) then x − y = t is even, but
y −x = −t is also even, hence y ≡ x (mod 2).
3. Transitive: assume x ≡ y (mod 2) and y ≡ z (mod 2). Then
x−y = t and y−z = u are even.

From here, x−z = (x−y)+(y −z) = t+u is also even, hence x ≡ z (mod 2).

As congruence modulo 2 is reflexive,symmetric and transitive it is an equivalence relation.

Thanks and Regards

Shaik Aasif

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