   Chapter 1.7, Problem 3E

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# a. Let R be the equivalence relation defined on Z in Example 2, and write out the elements of the equivalence class [ 3 ] .b. Let R be the equivalence relation “congruence modulo 4” that is defined on Z in Example 4. For this R , list five members of equivalence class [ 7 ] .

(a)

To determine

The elements of the equivalence class  for the relation R.

Explanation

Given Information:

The relation R defined on set of all integers Z as xRy if and only if |x|=|y|.

Explanation:

Here, xRy|x|=|y|.

First of all, clearly |3|=|3| and |3|=3, so 3 and 3. To prove that ={3,3}, we will prove that for every xZ, if x3 and x3, then x, or, more precisely, 3Rx.

Consider three cases;

Case I:

Let x>3. Then x is positive, so |x|=x, and |x|=x> 3=|3|, so |3|<|x|, which means that |3||x|. Finally, 3Rx follows from the definition of R

(b)

To determine

The elements of the equivalence class  for relation R.

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