Definition:Preimage/Mapping/Element

Definition
Let $S$ and $T$ be sets

Let $f: S \to T$ be a mapping.

Let $f^{-1} \subseteq T \times S$ be the inverse of $f$, defined as:


 * $f^{-1} = \set {\tuple {t, s}: \map f s = t}$

Every $s \in S$ such that $\map f s = t$ is called a preimage of $t$.

The preimage of an element $t \in T$ is defined as:


 * $\map {f^{-1} } t := \set {s \in S: \map f s = t}$

This can also be expressed as:
 * $\map {f^{-1} } t := \set {s \in \Img {f^{-1} }: \tuple {t, s} \in f^{-1} }$

That is, the preimage of $t$ under $f$ is the image of $t$ under $f^{-1}$.

Also known as
The preimage of an element is also known as its inverse image.

In other contexts, this is called the fiber of $t$ (under $f$).

The UK English spelling of fiber is fibre.

The term argument is popular in certain branches of mathematics.

If $\tuple {x, y} \in f$, then $x$ is the argument (of $f$) which holds the value $y$.

In the context of computability theory, the following terms are frequently found:

If $\tuple {x, y} \in f$, then $x$ is often called the input of $f$ which produces the output $y$.

Also see

 * Definition:Domain of Mapping
 * Definition:Codomain of Mapping
 * Definition:Range


 * Definition:Image of Element under Mapping


 * Definition:Preimage of Relation


 * From Preimages All Exist iff Surjection, $\map {f^{-1} } t$ is guaranteed not to be empty $f$ is a surjection.


 * From the definition of an injection, if $\map {f^{-1} } t$ is not empty, then it is guaranteed to be a singleton $f$ is an injection.

Thus, while $f^{-1}$ is always a relation, it is not actually a mapping unless $f$ is a bijection.