I've come across challenge proof question in my discrete mathematics textbook that I'm trying to solve for practice but unfortunately it does not have a solution. Any help with a reasonable explanation or solution so that I can understand where to start and verify my work would be greatly appreciated:
Suppose that $\mathcal U$ is the universal set, and that $A$, $B$ and $C$ are three arbitrary sets of elements of $\mathcal U$. Prove that if $A - B \subseteq C$, then $A - C \subseteq B$.
Thank you!
Since $A \cap B^\complement \subset C$, we must have $C^\complement \subset (A \cap B^\complement)^\complement = A^\complement \cup B$. Therefore, $$A \cap C^\complement \subset A \cap (A^\complement \cup B) = A \cap B \subset B.$$