Equivalence relation- Equivalence Classes And Partitions

Question

I had the following question

A is a finite set and $R \subseteq A \times A$ is a equivalence relation.

Prove that $|A|$ is odd iff $|R|$ is odd.

I am trying to find a general formula for this question, Because I think it is true to a even set too. Anyone have an idea?

Thanks for the help

Answer 1

$R$ contains the set $Q=\{(x,x); x\in A\}$ which has as many elements as $A$. Try proving that $R\setminus Q$ has an even number of elements.

Answer 2

HINT: If $R_i$ are the equivalence classes of $R$, then $|R|=\sum|R_i|^2$ and $|A|=\sum|R_i|$. Look at both sums $\bmod 2$.