Definition – Function. – Math-reference.net

Contents
1. Definition 1 – Set theoretical definition
2. Notation

Definition 1 – Set theoretical definition

In set theory and analysis (a branch of Mathematics similar to calculus, but deriving results from their original axioms; ie, more rigorous). Let $A$ and $B$ be sets, and consider their cross product, $A\times B$ . A function is any subset of $A\times B$ such that, for all $a\in{A}$ and $b,b'\in{B}$ , if $(a, b)$ and $(a, b')$ are both elements of $f$ , then $b=b'$ .

Notation

There are several ways to write functions. One way is to write $f:A\rightarrow{B}$ , which highlights the fact that $f$ takes values of $A$ , and outputs values of $B$ . Importantly, using this notation, we mean that $f$ takes all values of $A$ , but not necessarily all elements of $B$ are output by the function.