Let `R` be the relation defined on the set `A={1,\ 2,\ 3,\ 4,\ 5,\ 6,\ 7}` by `R={(a ,\ b):` both `a` and `b` are either odd or even}. Show that `R` is an equivalence relation. Further, show that all the elements of the subset {1, 3, 5, 7} are related to each other and all the elements of the subset {2, 4, 6} are related to each other, but no element of the subset {1, 3, 5, 7} is related to any element of the subset {2, 4, 6}.

Explanation

To solve the problem, we will follow these steps:

### Step 1: Define the relation
The relation \( R \) is defined on the set \( A = \{1, 2, 3, 4, 5, 6, 7\} \) as follows:
\[ R = \{(a, b) : a \text{ and } b \text{ are both odd or both even}\} \]

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