Find the equivalence class of this relation!

Question

I am having the following relation with the set A and B:

$$ (x_1, y_1) \sim_{A\times B} (x_2, y_2) \iff\; x_1 \sim_A x_2\ \;\land\; \; y_1 \sim_B \; y_2 $$

I haved already proved, that it is a equivalence relation. Now I search the equivalence class for $$\sim_{A\times B}$$

I know the definition of an equivalence class, $$ [x]:=(y\in A\mid x\sim y) $$

$$ ⟨x′,y′⟩∈[⟨x,y⟩]_{A×B}\ \textrm{ if and only if }\ x′∈[x]_A\ \textrm{and}\ y′∈[y]_B$$ Now I need a way to describe this: $${⟨x,y⟩:x∈X\ \textrm{and}\ y∈Y}$$

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What is the amount of the equivalence classes of this relation? How can I write it down?

Answer 1

You just need to use the Cartesian product. Then you can write: \begin{align*} [x,y]_{A\times B} &= \{(x',y') \in A\times B\ |\ (x',y')\sim_{A\times B} (x,y)\}\\ &= \{(x',y') \in A\times B\ |\ x' \in [x]_A \quad \text{and}\quad y'\in [y]_B\}\\ &= \{(x',y') \in [x]_A\times [y]_B\}\\ &= [x]_A\times [y]_B \end{align*}

Now, say we have $n_A$ different equivalence classes in $A$ and $n_B$ different equivalence classes in $B$. Then the number of different equivalence classes in $A\times B$ is $n_A\cdot n_B$.