How would you go about proving this:
× ⊆ × . Prove ⊆ .
I said, because S * T is a subset of T * W then every element of S * T must exist in T * W but, that can only happen if S = T and T = W. Knowing that, S = W and thus, S is a subset of W.
Is this direct proof the right way to prove this?
$$ S\times T \subseteq T\times W \implies S\subseteq T \text { and } T\subseteq W $$
Transitivity of inclusion implies $$S\subseteq W$$
It's the right idea, but you can't conclude, from $A \times B \subseteq C \times D$, that $A = C$ and $B =D$, only that $A \subseteq C$ and $B \subseteq D$. Therefore, from $S \times T \subseteq T \times W$, we see that $S \subseteq T \subseteq W \implies S \subseteq W$.
