Show that the relation `R` on the set `A={1,\ 2,\ 3,\ 4,\ 5}` , given by `R={(a ,\ b):|a-b|` is even }, is an equivalence relation. Show that all the elements of {1, 3, 5} are related to each other and all the elements of {2, 4} are related to each other. But, no element of {1, 3, 5} is related to any element of {2, 4}.

Explanation

To show that the relation \( R \) on the set \( A = \{1, 2, 3, 4, 5\} \), defined by \( R = \{(a, b) : |a - b| \text{ is even}\} \), is an equivalence relation, we need to prove that it is reflexive, symmetric, and transitive.

### Step 1: Prove Reflexivity
A relation is reflexive if every element is related to itself.

For any \( a \in A \):
\[
|a - a| = |0| = 0
...