# Universal set proof [discrete mathematics]

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, $$A \cap C^\complement \subset A \cap (A^\complement \cup B) = A \cap B \subset B.$$
• $B^\complement$ denotes the complement of $B$ (in the universal space). So, $A - B = A \cap B^\complement$. – Gary Moon Mar 16 '19 at 22:18