# Definition:Preimage/Mapping/Mapping

## Definition

Let $f: S \to T$ be 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

Consistently with the definition as the image of $T$ under $f$, $f^{-1} \sqbrk T$ can also be used instead of $\Preimg f$.

## Technical Note

The $\LaTeX$ code for $\Preimg {f}$ is \Preimg {f} .

When the argument is a single character, it is usual to omit the braces:

\Preimg f