If $B \subseteq C$, then $(A-C) \subseteq (A-B)$ (by element method)

Question

I am supposed to prove that if $B\subseteq C$, then $(A-C) \subseteq (A-B)$ by the element method, but I am not really sure where to start.

I believe that you could let $x \in (A-C)$ and you are supposed to show that $x \in (A-B)$, but I'm not sure how to go from Point A to Point B.

Any help would be much appreciated. Thank you!

Answer 1

Hint: $x\in A-C$ if and only if $x\in A$ and $x\not\in C$. You want to show that $x\in A-B$, i.e. $x\in A$ and $x\not\in B$. $x\in A$ should be obvious, so it remains to show that $x\not\in B$. Why is that?

Answer 2

$B \subset C$ :

$x \in B$ implies $x \in C$ or equivalently

$\star):$ $y \not \in C$ implies $y \not \in B.$

Show $A-C \subset A-B:$

$y \in A-C$ implies

$y \not \in C$ and $y \in A;$

By $\star)$: $y \not \in C$ implies $y \not \in B.$

Thus we have $y \not \in B$ and $y \in A$, or

$ y \in A-B.$