Sorry this is such a basic question, but I'm studying all on my lonesome.

I'm struggling with the function/mapping definition of binary operations. I get the idea that if $\displaystyle a,b \in A$ and $\displaystyle a \star b \in A$, then $\displaystyle \star $ is a binary operation. How does $\displaystyle f: A \times A \rightarrow A$ mean the same thing? It seems like if A were a set of individual numbers (such as the set $\displaystyle \mathbb{Z}$), then f would change A to a set of ordered pairs made from elements of A. Also, is f the binary operator in the definition?