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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 ( A ) be the power set of the nonempty set A , and let C denote a fixed subset of A .

Define R on ( A ) by x R y if and only if x C = y C . Prove that R is an equivalence relation on ( A ) .

To determine

To prove: R is an equivalence relation on (A) where (A) is the power set of the nonempty set A, and C is a fixed subset of A and R is a relation defined on (A) by xRy if and only if xC=yC.

Explanation

Formula Used:

A relation R on a nonempty set A is equivalence if the following properties are satisfied:

(1) xRx for all xA. (Reflexive Property)

(2) If xRy, then yRx. (Symmetric Property)

(3) If xRy and yRz, then xRz. (Transitive Property)

Proof:

Let x,y,z(A).

1. xRx,sincexC=xC.

So, R is reflexive.

2

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