Critique my proof on correctness, structure, etc.
Proof. Let $p = (x,y) \in (A \times B) \cup (C \times D)$. Thus, $(x, y) \in (A \times B)$ or $(x, y) \in (C \times D)$.
Case #1
Suppose $(x, y) \in (A \times B)$. By definition of cartesian product, $x \in A$ and $y \in B$. It follows that $x \in (A \cup C)$ and $y \in (B \cup D)$ by definition of union.
Case #2
Suppose $(x, y) \in (C \times D)$. By definition of cartesian product, $x \in C$ and $y \in D$. It follows that $x \in A \cup C$ and $y \in C \cup D$ by definition of union.
$\therefore$ Because $(x, y)$ is an arbitrary element of $(A \times B) \cup (C \times D)$, $(A \times B) \cup (C \times D) \subseteq (A \cup C) \times (B \cup D)$.
Best Answer
This proof is very tight and if I was grading it I'd give it full credit.
If you want to be more concise you could say that since the union operator is commutative, that proving only case one is necessary since case two is the same argument just relabeling the sets.
Something along the lines of
However, perhaps I should not be encouraging people to be lazy in proofs like me