   Chapter 1.7, Problem 6E

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# In Exercises 6 − 10 , a relation R is defined on the set Z of all integers, In each case, prove that R is an equivalence relation. Find the distinct equivalence classes of R and least four members of each. x R y if and only if x 2 + y 2 is a multiple of 2 .

To determine

To prove: The relation defined by xRy if and only if x2+y2 is a multiple of 2 on the set of all integers Z is an equivalence relation and also find the distinct equivalence classes of R and list at least four members of each.

Explanation

Formula Used:

The formula used in this proof is x2+y2=2k.

Proof:

We need to prove that R is reflexive, symmetric and transitive.

For reflexive relation xRx must hold good, so, x2+x2=2x2, which is a multiple of 2. Thus, relation R is reflexive.

For symmetric relation, xRyx2+y2=2k for some kZ and yRxy2+x2 as addition follows the commutative law x2+y2=y2+x2 and both will be multiple of 2.

Thus, the relation R is symmetric.

For, transitive relation, xRyx2+y2=2k and yRzy2+z2=2l for some k,lZ.

Now x2+z2=2(k+ly2) which is divisible by 2, Thus, the relation xRy is transitive

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