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Show that the relation R in the set A =...

Show that the relation R in the set `A = {1, 2, 3, 4, 5}`given by `R = {(a , b) : |a - b| is e v e n}`, 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}`.

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To show that the relation \( R \) in 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 verify three properties: reflexivity, symmetry, and transitivity. ### Step 1: Check Reflexivity A relation is reflexive if for every element \( a \in A \), the pair \( (a, a) \) belongs to \( R \). - For any \( a \in A \): \[ |a - a| = |0| = 0 ...
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