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

Answer 1

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.