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I'm trying to practice proof writing, and found the following text question:
For all sets A,B,C:
$ (A \cup B) \cap C \subseteq A \cup (B \cap C)$
The first step I was thinking of showing is that:
$ (A \cup B) \cap C = (A \cap C) \cup (B\cap C)$
or the subset is the sliver only where A and B intersect C.
I'm kind of stuck on where to go from here. I know the statement is true as the superset contains ALL of A, and the parts where C and B intersect, but I'm not really sure how to prove this. Any tips?