# Definition:Preimage

## Relation

Let $\mathcal R \subseteq S \times T$ be a relation.

Let $\mathcal R^{-1} \subseteq T \times S$ be the inverse relation to $\mathcal R$, defined as:

$\mathcal R^{-1} = \left\{{\left({t, s}\right): \left({s, t}\right) \in \mathcal R}\right\}$

### Preimage of Element

Every $s \in S$ such that $\left({s, t}\right) \in \mathcal R$ is called a preimage of $t$.

In some contexts, it is not individual elements that are important, but all elements of $S$ which are of interest.

Thus the preimage of $t \in T$ is defined as:

$\mathcal R^{-1} \left ({t}\right) := \left\{{s \in S: \left({s, t}\right) \in \mathcal R}\right\}$

This can also be written:

$\mathcal R^{-1} \left ({t}\right) := \left\{{s \in \operatorname{Im} \left({\mathcal R^{-1}}\right): \left({t, s}\right) \in \mathcal R^{-1}}\right\}$

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

### Preimage of Subset

Let $Y \subseteq T$.

The preimage of $Y$ under $\mathcal R$ is defined as:

$\mathcal R^{-1} \left [{Y}\right] := \left\{{s \in S: \exists y \in Y: \left({s, y}\right) \in \mathcal R}\right\}$

That is, the preimage of $Y$ under $\mathcal R$ is the image of $Y$ under $\mathcal R^{-1}$.

If no element of $Y$ has a preimage, then $\mathcal R^{-1} \left [{Y}\right] = \varnothing$.

### Preimage of Relation

The preimage of $\mathcal R \subseteq S \times T$ is:

$\Preimg {\mathcal R} := \mathcal R^{-1} \left [{T}\right] = \set {s \in S: \exists t \in T: \tuple {s, t} \in \mathcal R}$

## Mapping

$\mathcal R$ can also be (and usually is in this context) a mapping.

Exactly the same notation and terminology concerning the concept of the preimage applies to the inverse of a mapping.

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

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

$f^{-1} = \left\{{\left({t, s}\right): f \left({s}\right) = t}\right\}$

### Preimage of Element

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

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

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

This can also be expressed as:

$f^{-1} \paren t := \set {s \in \Img {f^{-1} }: \tuple {t, s} \in f^{-1} }$

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

### Preimage of Subset

Let $Y \subseteq T$.

The preimage of $Y$ under $f$ is defined as:

$f^{-1} \sqbrk Y := \set {s \in S: \exists y \in Y: \map f s = y}$

That is, the preimage of $Y$ under $f$ is the image of $Y$ under $f^{-1}$, where $f^{-1}$ can be considered as a relation.

If no element of $Y$ has a preimage, then $f^{-1} \sqbrk Y = \O$.

### Preimage of Mapping

The preimage of $f$ is defined as:

$\Preimg f := \set {s \in S: \exists t \in T: f \paren s = t}$

That is:

$\Preimg f := f^{-1} \sqbrk T$

where $f^{-1} \sqbrk T$ is the image of $T$ under $f^{-1}$.

In this context, $f^{-1} \subseteq T \times S$ is the the inverse of $f$.

It is a relation but not necessarily itself a mapping.

## Also known as

Some sources spell preimage with a hyphen and write pre-image.

A preimage is also known as an inverse image.