# How to show that a relation is an equivalence relation given a defined relation

I understand for the most part the conceptual aspects of an equivalence relation. A relation is considered a equivalence relation if it satisfies reflexive, symmetric and transitive properties but Im having trouble working with this on paper.

For example, Given a relation R defined on the integers by aRb <=> a+b is even, show that this relation is an equivalence relation.

So far my approach is. Reflexive, aRa <=> a+a is even Symmetric, if bRa <=> b+a is even Transitive, if aRb and bRc then aRc <=>a+c is even.

But after that i am stuck.

• You have the right definitions, now you need to apply them. For example, for reflexivity, as you wrote aRa <=> a+a is even. If a is an integer, can you see why a+a will always be even? The other properties are done similarly. – pwerth Mar 5 '15 at 4:03

## 1 Answer

Just apply the definition :

• reflexivity : $a+a = 2 a$ is even, so $aRa$

• symetric : if $aRb$ then a+b is even, then b+a is even then $bRa$

• transitive : if $aRb$ and $bRc$ then a+b is even and b+c is even, then a+2b+c is even then a+c is even (because 2b is even), then $aRc$