Math 310-01/02

11-3-2017

1. Prove that

Suppose . Then for ,

Hence, . Since this is true for all , it follows that .

Suppose . Then for all . Hence,

I claim that . Suppose on the contrary that . Now , so for some

This contradicts the fact that for all .

Hence, , and so .

Therefore, .

2. An equivalence relation is defined on the set by means that a and b differ by an (integer) multiple of 5. List the elements of the equivalence classes for this relation.

3. For each relation, check *each* of the axioms for an
equivalence relation. For each axiom, if the axiom holds, prove it.
If the axiom doesn't hold, give a specific counterexample.

(a) on means that or .

(b) on means that .

(c) on means .

(a) For all , , so . Hence, the relation is reflexive.

Suppose . There are two cases. First, if , Then , so . Second, if , then , so . Thus, the relation is symmetric.

I have , since . I have , since . However, , so . Therefore, the relation is not transitive.

(b) , so . Therefore, the relation is not reflexive.

Suppose . Then , so . Therefore, . Thus, is symmetric.

Since , I have . Since , I have . However, , so . Therefore, the relation is not transitive.

(c) Since , I have . Thus, is reflexive.

If , then . Hence, , so . Thus, is symmetric.

If and , then

Therefore, , so is transitive.

It follows that is an equivalence relation.

*The earth keeps some vibrations going There in your heart, and
that is you.* - *Edgar Lee Masters*

Copyright 2017 by Bruce Ikenaga