How can I prove that for any $A, B$ if $A\subseteq B$ and $B\subseteq C$, then $(C-A)\cup (B-A)\subseteq C$?
I've been working on this question and I haven't really made much progress with it. I know that I can rewrite it as $(C \cap A^c) \cup (B\cap A^c)$. I'm pretty sure that if $A \subseteq B$ and $B \subseteq C$ then $A \subseteq C$. If this is the case then wouldn't $(C \cap A^c) = \emptyset$ and $(B \cap A^c) = \emptyset$ or am not understanding something with set theory? Thank you for the help.