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 $(A\circ B)\circ C=A\circ(B\circ C)$?

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