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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!

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Since $A \cap B^\complement \subset C$, we must have $C^\complement \subset (A \cap B^\complement)^\complement = A^\complement \cup B$. Therefore, $$A \cap C^\complement \subset A \cap (A^\complement \cup B) = A \cap B \subset B.$$

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  • $\begingroup$ I'm not quite sure I understand your notation, what does BC mean? $\endgroup$ – Joe Biden Mar 16 '19 at 22:10
  • $\begingroup$ $B^\complement$ denotes the complement of $B$ (in the universal space). So, $A - B = A \cap B^\complement$. $\endgroup$ – Gary Moon Mar 16 '19 at 22:18

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