How many equivalence classes does a set have

Question

For $A = \{a, b\}$ and $B = A^* × A^*$ ($A^*$ is the set of all words (of all lengths))

In my opinion, because $B$ is reflexive, symmetric and transitive $(aa, bb)$. It is an equivalence relation, the equivalence classes should be one, because all elements are related to each other?

Am I correct?

Answer 1

Yes you're right. Two elements belong to the same equivalence class if and only if they are equivalent. If you take any two elements $w_1,w_2\in A^*$, we have $(w_1,w_2)\in A^*\times A^*=B$, which shows that they are both equivalent under the equivalence relation $B$.