Question:
Prove $(A \times C) \cup (B \times D) \subset (A \cup B) \times (C \cup D)$ where $A,B,C,D$ are non empty sets.
My Attempt:
$$(x \in A \land x \in C) \lor (x \in B \land x \in D) \implies (x \in A \lor x \in B) \land (x\in C \lor x\in D) $$
That is as far as I got.
Question:
Could someone please tell me if what I have done is correct and/or show what needs to be done?
You have the right idea, but remember that the Cartesian product of two sets is a set of ordered pairs.
Thus you seek to show, for any $x$ and any $y$, that: $${(x{\in}A\wedge y{\in}C)\vee(x{\in}B\wedge y{\in}D))}~\to~ {(x{\in}A\vee x{\in}B)\wedge(y{\in}C\vee y{\in}D)}$$
Now go show that.