Preimage of a function

The set of inputs mapped into a target subset T⊆B, denoted f^{-1}(T)

Preimage of a function

Let $f\colon A\to B$ be a function and let $T\subseteq B$ be a subset . The preimage (or inverse image) of $T$ under $f$ is

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

Preimages are defined for arbitrary functions (not only bijections ) and are fundamental in pulling back structure along maps.

Examples:

If $f(x)=x^2$ on $\mathbb{R}$ and $T=[0,1]$ , then $f^{-1}(T)=[-1,1]$ .
If $f\colon\mathbb{Z}\to\mathbb{Z}$ is $f(n)=2n$ and $T=\{0\}$ , then $f^{-1}(T)=\{0\}$ .