# A set operation is associative if and only if the binary operator defining it is associative

Let $$\star$$ be a binary operation on $$\Bbb{N}$$. For all $$A,B \subseteq\Bbb{N}$$, $$\circ$$ is defined as:

$$A \circ B = \left\{a \star b \mid a \in A \wedge b \in B\right\}$$

I'm trying to prove that $$A \circ B$$ is associative if and only if $$a \star b$$ is associative. Proving the direction $$a \star b$$ $$\rightarrow$$ $$A \circ B$$ is easy enough, but I'm having trouble with the reverse direction. Any suggestions?

Hint: If $$A=\{a\}$$, $$B=\{b\}$$, and $$C=\{c\}$$, what can you deduce from the fact that $$(A\circ B)\circ C=A\circ(B\circ C)$$?