Functions:Definition

Example of an arrow diagram of a function $f:A\to B$

A function (or "mapping") is a relationship between two sets $A$ and $B$ that maps each input $a\in A$ to exactly one output $b\in B$. A function $f$ that maps elements of the set $A$ to elements in the set $B$ is denoted as $f:A\to B$, where $A$ is the domain of $f$ and $B$ is the codomain. We can also think of a function $f:A\to B$ as a set of ordered pairs $(a,b)$, $a\in A$ and $b\in B$, such that each element $a$ is paired with exactly one element $b$. If a function $f:A\to B$ maps an input $a$ to an output $b$, we can write that $f(a)=b$. For finite, reasonably small sets, we can depict a function graphically (see image).

Note: a function cannot map one input to more than one output, but it can map more than one input to the same output. For example, let $A=\{a_{1},a_{2}\}$ and $B=\{b_{1},b_{2}\}$, $a_{1}\neq a_{2}$ and $b_{1}\neq b_{2}$. If $f:A\to B$ is a relation such that $f(a_{1})=b_{1}$ and $f(a_{1})=b_{2}$, then $f$ is NOT a function. However, a relation $g:A\to B$ such that $g(a_{1})=b_{1}$ and $g(a_{2})=b_{1}$ IS a valid function.

Examples:

• Let $A=\{1,2,3,4,5\}$ and $B=\{-1,3,4\}$, and let $f:A\to B$ such that $f(1)=-1$, $f(2)=3$, $f(3)=4$, $f(4)=3$, and $f(5)=-1$. Since each element of the domain maps to exactly one element (that is, there is no $f(a)=b_{1}$ and $f(a)=b_{2}$ such that $b_{1}\neq b_{2}$), $f$ is a function.

• For sets $A$ and $B$ in the previous example, let $g:A\to B$ be a relation such that $g(1)=1$, $g(1)=3$, $g(2)=4$, $g(4)=4$, and $g(5)=4$. Since $g$ maps the input $1$ to two distinct outputs, this relation is NOT a valid function.

• Let $f:\mathbb {N} \to \mathbb {N}$ such that $f(n)=n+1$. This is a function, since each $n\in \mathbb {N}$ maps to exactly one element $n+1\in \mathbb {N}$.

• Let $g:\mathbb {N} \to \mathbb {Z}$ such that $|g(n)|=n$. This is not a valid function, since for $n\in \mathbb {N}$, $g(n)$ can equal both $n$ and $-n$, and $n\neq -n$ for $n\neq 0$.

Resources

• Course Textbook, pages 129-140
• Definition of Functions, Mathematics LibreTexts