0
$\begingroup$

I'm not even sure where to start or how to get to proving this. So far what I did isn't much, but

Assume $A \subseteq B$ and let $x,y \in A \times C$, then $x \in A$ and $y \in C$.

Do I then go on to claim that $z,y \in B \times C$ since for the first part we already stated that $y \in C$? If so, where do I go from there?

  • 0
    Focus on what you are asked to prove, that $A\times C \subseteq B\times C$. Notice the definition of $A\times C$. It is a set of ordered pairs. – 2017-02-17

1 Answers 1

1

Other than that you should be consistent with $x$ and $z$, and your pairs should go in parentheses to make it clear that they're pairs, that's right. We suppose that $(x, y) \in A \times C$. In that case, $x \in A$ and $y \in C$. But because $A \subseteq B$, it must also be the case that $x \in B$. If $x \in B$ and $y \in C$, then it must be true that $(x, y) \in B \times C$. Therefore $A \times C \subseteq B \times C$.