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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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Suppose { A λ } , λ £ , represents a partition of the nonempty set A. Define R on A by x R y if and only if there is a subset A λ such that x A λ and y A λ . Prove that R is an equivalence relation on A and that the equivalence classes of R are the subsets A λ .

To determine

To prove: R is an equivalence relation on A and that the equivalence classes of R are the subsets Aλ.

Explanation

Given Information:

Suppose {Aλ}, λ£, represents a partition of the nonempty set A. R is defined on A by xRy if and only if there is a subset Aλ such that xAλandyAλ.

Proof:

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

1. Reflexive Property: xRxforallxA.

2. Symmetric Property: If xRy, then yRx.

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

Consider the given information.

Suppose {Aλ}, λ£, represents a partition of the nonempty set A. R is defined on A by xRy if and only if there is a subset Aλ such that xAλandyAλ.

1. xRx, since xAxλ£Aλ.

2. xRyx,yAλforsomeλy,xAλforsomeλyRx

3

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