Definition:Preimage/Mapping

Definition

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 an Element

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

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

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

This can also be written:

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

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

Preimage of a Subset

Let $Y \subseteq T$.

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

$f^{-1} \left ({Y}\right) := \left\{{s \in S: \exists y \in Y: f \left({s}\right) = y}\right\}$

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 $f^{-1} \left ({Y}\right) = \varnothing$.

Preimage of a Mapping

The preimage of $f$ is defined as:

$\operatorname{Im}^{-1} \left ({f}\right) := \left\{{s \in S: \exists t \in T: f \left({s}\right) = t}\right\}$

That is:

$\operatorname{Im}^{-1} \left ({f}\right) := f^{-1} \left ({T}\right)$

where $f^{-1} \left ({T}\right)$ is the image of $T$ under $f$.

Also known as

A preimage is also known as an inverse image.

Also see

• Preimage of a Relation

• Domain
• Codomain
• Range

• Image

Sources

• René L. Schilling: Measures, Integrals and Martingales (2005)... (previous)... (next) $\S 2$