Thread: Equivalence relation - Congruence modulo

2. Re: Equivalence relation - Congruence modulo

What have you tried so far? You should prove by definition that the given relation is reflexive, symmetric and transitive. Can you do that?

3. Re: Equivalence relation - Congruence modulo

Thank you. I am familiar with the properties reflexive, symmetric and transitive, but not when it comes to modulo. I have never seen it before, and simply do not know where to start.

4. Re: Equivalence relation - Congruence modulo

The relation $\equiv_5$ is defined as $\forall x,y \in \mathbb{S}_7: x\equiv_5 y \Leftrightarrow x \mod 5 = y \mod 5$
To check if the relation is reflexive you have to check $\forall x \in \mathbb{S}_7: x \equiv_5 x$ which is true because $x \mod 5 = x \mod 5$.

Can you check the symmetric and transitive property now?

5. Re: Equivalence relation - Congruence modulo

Originally Posted by aprilrocks92
I have a different but equidistant way of describing that relation.
Say $x\mathcal{R}y$ if and only if $x~\&~y$ have the same remainder when divided by $5$

Thus it should be clear that $2\mathcal{R}7$.

The three needed properties are easily checked.

There are five equivalence classes.