   Chapter 1.7, Problem 27E

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# Prove Theorem 1.40: If R is an equivalence relation on the nonempty set A , then the distinct equivalence classes of R form a partition of A .

To determine

To prove: If R is an equivalence relation on the nonempty set A, then distinct equivalence classes of R forms the partitions of A.

Explanation

Formula Used:

i) Let {Aλ},λL be a collection of subsets of the nonempty set A. Then {Aλ},λL is a partition of A if all these conditions are satisfied:

Each Aλ is nonempty.

A=λL{Aλ}.

If AαAβ then Aα=Aβ.

ii) Let R be an equivalence relation on nonempty set A. For each aA, the set [a]={xA|xRa} is called equivalence class containing a.

Proof:

Suppose that R is an equivalence relation on the nonempty set A.

For each aA

Equivalence class of a is [a]={xA|xRa}.

Thus a[a] for all aA.

Hence [a] is nonempty for all aA …….. (1)

If [a1][a2] for a1,a2A

Thus there exists c[a1][a2]

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