Equivalence relation and distinct classes

Question

A relation $R$ is defined on $\Bbb N$ by $aRb$ if $a^2+b^2$ is even.

a) Prove $R$ is an equivalence relation.

b) Determine the distinct equivalence classes.

I am having trouble with the transitive part of the proof and the distinct classes.

Answer 1

It is just congruence modulo 2. This is an equivalence relation by a previous result in your book.

Answer 2

A common trick to prove transitivity when working with relations involving "$+$" or "$-$" is to add them together. In this case, suppose $aRb$ and $bRc$, so $a^{2}+b^{2}$ and $b^{2}+c^{2}$ are even. If we add these together we find that $a^{2}+2b^{2}+c^{2}$ is even. Thus, since $2b^{2}$ is even, we see that $a^{2}+c^{2}$ is even. In other words, we have shown $aRc$. This proves transitivity.

For the distinct classes, just observe that $aRb$ if and only if $a,b$ are both odd or both even (i.e. $a,b$ are congruent modulo $2$). Thus the two distinct equivalence classes are just the even numbers and the odd numbers.