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

Given `A = {1, 2, 3, 4, 5, 6, 7} `


and `R = {(a, b) :` both` a` and `b` are either odd or even number}


Therefore,


`R = {(1, 1), (1, 3), (1, 5), (1, 7), (3, 3), (3, 5), (3, 7), (5, 5), (5, 7), (7, 7), (7, 5), (7, 3), (5, 3), (7, 1), `

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