   Chapter 1.7, Problem 7E

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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 5 .

To determine

To prove:

The relation defined by xRy if and only if x2y2 is a multiple of 5 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 x2y2=5k.

Proof:

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

For reflexive relation xRx must hold good, so, x2x2=0, which is a multiple of 5. Thus, relation R is reflexive.

For symmetric relation, xRyx2y2=5k for some kZ and yRxy2x2 as y2x2=5k which is also a multiple of 5.

Thus, the relation R is symmetric.

For, transitive relation, xRyx2y2=5k and yRzy2z2=5l for some k,lZ. Adding both relations, we get

x2z2=5(k+l) which is divisible by 5.

Thus, the relation xRy is transitive.

Hence, the relation xRy is an equivalence relation.

Consider onto equivalence classes.

First note that xRyx2y2 is multiple of 5.

Since x2y2=(x+y)(xy), we conclude that

xRy(x+y)(xy)=5k

For some kZ. Since 5 divides the right side, it divides the left side of the equality as well. Since 5 is prime. It either divides (x+y) or (xy)

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