Question

Let R be the equivalence relation in the set Z of integers given by R={(a,b):2 divides a-b}.Write the equivalence class [0].

Answer 1

An equivalence relation is a relation which is reflexive, symmetric and also transitive. Hence, if R possesses all these 3 properties, then R is called an equivalence relation.
Example : Property of congruence on a set of triangles in Euclidean plane geometry.
i) Reflexive: triangle A is congruent to itself.
ii) Symmetric :If triangle A is congruent to triangle B, it follows that triangle B is congruent to triangle A.
iii) Transitive : If triangle A is congruent to triangle B & triangle B is congruent to triangle C, then it is obvious that triangle A is congruent to triangle C.
Hence, congruence is an equivalence relations.

Now your problem:

An important property of equivalence relation is that it divides the set into pairwise disjoint subsets which are called equivalence classes.
Here, the meaning of equivalence class [0] means set of pairs of a, b which given value a - b = 0, Hence, all the pairs of +ve and -ve same numbers will belong to this class.