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Equivalence relation on set of ordered pairs

Let S be the set of ordered pairs of positive integers, let z = (5,8), and define R so that (x1, x2) R (y1, y2) means that x1 + y2 = y1 + x2.

Show that the given relation R is an equivalence relation on the set S. Then describe the equivalence class containing the given element z of S, and determine the number of distinct equivalence classes of R.

Solution Preview

To show that R is an equivalence relation on set S, we need to show that R is reflexive, symmetric, and transitive.

1. To show that R is reflexive, we need to show the following:

For every ordered pair (z1, z2) of positive integers, (z1, z2) R (z1, z2).

Let (z1, z2) be an ordered pair of positive integers, and think of setting x1, x2, y1, and y2 in the definition of R to z1, z2, z1, and z2, respectively.

Then what we need to show is that z1 + z2 = z1 + z2.

Well, the expression on the right-hand side of the equals sign (z1 + z2) is identical to the expression on the left-hand side (z1 + z2), so it's true that z1 + z2 = z1 + z2; hence it's true that (z1, z2) R (z1, z2). Thus R is reflexive.

2. To show that R is symmetric, we need to show the following:

If (z1, z2) R (z3, z4), then (z3, z4) R (z1, z2).

So let (z1, z2) and (z3, z4) be ordered pairs of positive integers, and suppose that (z1, z2) R (z3, z4). What that means is that

z1 + z4 = z3 + z2 (call this Eq. (1))

What we need to show is that z3 + z2 = z1 + z4

Well, note that the expression on the left-hand side of Eq. (1) (namely, z1 + z4) is identical to the expression on the right-hand side of the equation z3 + z2 = z1 + z4, and that the expression on the right-hand side of Eq. (1) (namely, z3 + z2) is identical to the ...

Solution Summary

A detailed proof of the fact that the given relation R on the set S of all ordered pairs of positive integers is an equivalence relation is provided. The equivalence class containing the given element z is described. A detailed determination of the number of distinct equivalence classes of R is given, as are examples of elements of several different equivalence classes.

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