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 .
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.
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