Let A, B and C be nonempty sets. Prove that AxC is a subset of BxC if and only if A is a subset of B

Question

Let  A, B and C be nonempty sets. Prove that AxC is a subset of BxC if and only if A is a subset of B

anonymous · Answer

Let $a\in A$, $b\in B$, and $c\in C$. For a proof of the left-to-right statement, let $A\times C\subseteq B\times C$.

Recalling that $X\times Y$ is the set of 2-tuples $(x,y)$ such that $x\in X$ and $x\in Y$, this means that having $A\times C\subseteq B\times C$ means $(a,c)\in A\times C$ as well as $(a,c)\in B\times C$, which by definition entails that $a\in B$. So if $a\in A$ implies $a\in B$, this means $A\subseteq B$.

Try the other direction for yourself.