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Let R be the relation defined in the set...

Let R be the relation defined in 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}.

Text Solution

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Given, A = {3,5}, B = {7,11}.
Now, `R={(a,b):ainA,binBanda-b" is even"}`
`={(3,7),(3,11),(5,7),(5,11)}`
Also, `AxxB={(3,7),(3,11),(5,7),(5,11)}`
Clearly, `R = A xx B`
Hence, R is an universal relation from A to B.
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