Show that the relation `R` on the set `Z` of integers, given by `R={(a ,\ b):2` divides `a-b}` , is an equivalence relation.

Explanation

To show that the relation \( R \) on the set of integers \( \mathbb{Z} \), defined by \( R = \{(a, b) : 2 \text{ divides } (a - b)\} \), is an equivalence relation, we need to verify that it satisfies three properties: reflexivity, symmetry, and transitivity.

### Step 1: Check Reflexivity
A relation is reflexive if every element is related to itself. In mathematical terms, for all \( a \in \mathbb{Z} \), we need to show that \( (a, a) \in R \).

- Consider \( a - a = 0 \).
- Since \( 2 \) divides \( 0 \) (because \( 0 = 2 \times 0 \)), we have \( (a, a) \in R \).

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