I have a feeling that this isn't true, but I cannot for the life of me think of a counter example. I've tried thinking about the empty set, but then that leaves me with:
$ A \cup \emptyset = A \\A \cap \emptyset = \emptyset$
And so it's the case that $A \cap \emptyset \subset \ A \cup \emptyset $, which is the opposite of what I wanted. All other examples I've tried involved using sets A, B such that A=B. Any hints would be much appreciated!