   Chapter 1.7, Problem 3TFE

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# Label each of the following statements as either true or false.If R is an equivalence relation on a nonempty set A , then the distinct equivalence classes of R form a partition of A .

To determine

Whether the statement, “If R is an equivalence relation on the nonempty set A, then distinct equivalence classes of R forms the partitions of A ” is true or false.

Explanation

Consider the statement, “If R is an equivalence relation on the nonempty set A, then distinct equivalence classes of R forms the partitions of A

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]c[a1] and c[a2]

By using definition of equivalence class,

cRa1 and cRa2

Thus by using R is symmetric,

a1Rc and cRa2

By using R is transitive,

a1Ra2

T

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