Question

Decide for each of the following relations whether or not it is an equivalence relation. Give full reasons if it is an equivalence relation, give the equivalence classes.

A. Let a,b E Z. Define aRb if and only if (a/b)E Z.

B. Let a and b be integers. Define aRb if and only if3|(a-b) (is the congruence modulo 3 relation)

Answer 1

a)

The relation is reflexive: $a/a\isin Z$ ,

 non-symmetric: it is not always if $a/b\isin Z$, then $b/a\isin Z$

So, this is not equivalence relation.

b)

The relation is reflexive: $3|(a-a)=3|0$

symmetric: $3|(b-a)$

transitive: if $3|(a-b)$ and $3|(b-c)$ , then $3|(a-c)$

So, this is equivalence relation.

Equivalence classes: integer numbers divisible by 3