# Definition:Preimage

## Definition

### Relation

The preimage of $\RR \subseteq S \times T$ is:

$\Preimg \RR := \RR^{-1} \sqbrk T = \set {s \in S: \exists t \in T: \tuple {s, t} \in \RR}$

### Mapping

$\RR$ 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.

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.