Preimage - Knowlpedia

A preimage is the set of inputs that map into a given subset of the codomain: if $f:A\to B$ is a function and $T\subseteq B$ , then

f^{-1}(T)=\{a\in A: f(a)\in T\}\subseteq A.

Remarks

Unlike an inverse function, the preimage $f^{-1}(T)$ is defined for every function and every subset $T$ of the codomain. Preimages interact well with set operations such as union and intersection.

Examples

For $f:\mathbb{R}\to\mathbb{R}$ given by $f(x)=x^2$ , the preimage of $\{1\}$ is $f^{-1}(\{1\})=\{-1,1\}$ .
For $p:\mathbb{Z}\to\{0,1\}$ defined by parity, the preimage $p^{-1}(\{0\})$ is the set $\{2k:k\in\mathbb{Z}\}$ of even integers.