Let $R\subset X\times X, |X|=30$. Supposing there are only 3 distinct equivalence classes and all of these have the same amount of elements, find $|R|$.
I didn't get very far on this, I thought that because $R$ is reflexive all the pairs of the form $(a,a): a\in X$ are in $R$. Also that $3\mid |R|$ and that $30\leq |R| \leq 30^2=|X^2|$.
Do I have any more information I'm not seeing?
Each equivalence class has ten elements, and for any $a,b$ within the same equivalence class, $R$ has the element $(a,b)$. Hence each class contributes $10\cdot 10$ and ultimately we find that $|R|=300$.