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Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
ISBN: 9781285463230

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BuyFindarrow_forward

Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
ISBN: 9781285463230
Textbook Problem
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Let R be a relation defined on the set Z of all integers by x R y if and only if sum of x and y is odd. Decide whether or not R is an equivalence relation. Justify your decision.

To determine

Whether R is an equivalence relation or not.

Explanation

Given information:

R be a relation defined on the set of all integers by xRy if and only if the sum of x and y is odd.

Formula used:

Definition of equivalence relation:

A relation R on a non-empty set A is an equivalence relation if the following conditions are satisfied for arbitrary x,y,z in A:

1. Reflexive property: xRx for all xA.

2. Symmetric Property: If xRy, then yRx.

3. Transitive Property: If xRy and yRz, then xRz

Explanation:

1. Reflexive property:

For x

If xRxx+x=2x which is even.

Thus, sum of x and x is not odd.

Therefore the relation R is not reflexive.

2. Symmetric Property:

If xRy that is the sum of x and y is odd.

By using addition in is commutative,

Sum of x and y is equal to sum of y and x.

Hence the sum of y and x is also odd.

Thus yRx.

Therefore the relation R is symmetric.

3. Transitive Property:

If xRy and yRz

That is sum of x and y is odd and sum of y and z is odd

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