Equivalence relation help

Question

If $|A| = 30$ and the equivalence relation $R$ on $A$ partitions $A$ into (disjoint) equivalence classes $A_1$, $A_2$, and $A_3$, where $|A_1| = |A_2| = |A_3|$, then what is $|R|$?

Answer 1

We have $R = A_1 \times A_1 \cup A_2 \times A_2 \cup A_3 \times A_3$, so $|R|=|A_1\times A_1|+|A_2\times A_2|+|A_3\times A_3|$.

Answer 2

Hint: $$ R = \{(x_1,y_1):x_1,y_1\in A_1\} \cup \{(x_2,y_2):x_2,y_2\in A_2\} \cup \{(x_3,y_3):x_3,y_3\in A_3\}. $$

Count each of these separately, then add.

Answer 3

Hint: Note that for $j=1,2,3$, and any $x,y\in A_j,$ we have $(x,y)\in R.$ That means that $A_j\times A_j\subseteq R$ for $j=1,2,3$. In fact (why?), $$R=\bigcup_{j=1}^3A_j\times A_j,$$ and this union is disjoint. Can you take it from here?