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.

1. List the orderedpairs in the equivalencerelations produced by these partitions of {0,1,2,3,4,5}
a) {0}, {1,2}, {3,4,5}
b) {0,1}, {2,3}, {4,5}
c) {0,1,2}, {3,4,5}
d) {0}, {1}, {2}, {3}, {4}, {5}
2. Which of these collections of subsets are partitions of the set of integers?
a) the set of even integers and the set of

Relations
1. Let C = {2, 3, 4, 5} and D = {3, 4} and define a binary relation S from C to D as follows:
for all (x, y) C D, (x, y) S x y.
a) Write S as a set of orderedpairs.
b) Is 2 S 4? Is 4 S 3? Is (4, 4) S? Is (3, 2) S?
2. Let A = {3, 4, 5} and B = {4, 5, 6} and let S be the "divides" rel

Please see the attached.
Consider the following Hasse diagram of a partial ordering relation R on a set A:
(a) List the orderedpairs that belong to the relation. Keep in mind that a Hasse diagram is a graph of a partial ordering relation so it satisfies the three properties listed in number 5 part(b).
(b) Find the (Boo

Concerning discrete math, I am very confused as to the relationship between an equivalencerelation and an equivalence class.
I would very much appreciate it if someone could explain this relationship and give examples of each such that the relationship (or difference) is clear.

Please help with the following problem. Provide step by step.
Show that equality of integers is an equivalencerelation, that is show that equality of integers is reflexive, symmetric, and transitive. Recall two integers z=a--b, w=c--d, a, b, c, d belong to N (natural numbers) are equal if and only if a+d=b+c
**where a--b

Show that the following are equivalent:
(a) ~ is an equivalencerelation on a group G
(b) ~ is reflexive and, for all elements a, b, c of G: if a ~ b and b ~ c, then c ~ a.
See the attached file.

For m, n, in N define m~n if m^2 ? n^2 is a multiple of 3.
(a.) Show that ~ is an equivalencerelation on N.
(b.) List four elements in the equivalence class [0].
c) List four elements in the equivalence class [1].
(d.) Are there any more equivalence classes. Explain your answer.