Function

Given two sets $A$ and $B$, a function $f: A \rightarrow B$ takes an element of $a$ and maps to some element $f(a) \in B$ - importantly for every $a \in A$ there is only 1 element it maps to. Formally $f \subset A \times B$ such that for all $a \in A$ $\vert f \cap \{(a,b) \vert b \in B\}\vert = 1$. (We think of the set $f$ as $\{(a,f(a)) \vert a \in A\}$.)