Is it always true that

$(A\_1 \cup A\_2) \times (B\_1 \cup B\_2)=(A\_1\times B\_1) \cup (A\_2 \times B\_2)$?

I don't believe this is true. I have tried to draw pictures to help me get on the right path, but I think that the union makes this untrue. for example, if $a \in A\_1$ and $b \in B\_2$, then $(a,b)$ would not be in $(A\_1\times B\_1) \cup (A\_2 \times B\_2)$. Is this a correct assumption?