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:
- $f^{-1} = \set {\tuple {t, s}: \map f s = t}$
Preimage of Element
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 $f^{-1}$.
Preimage of Subset
Let $Y \subseteq T$.
The preimage of $Y$ under $f$ is defined as:
- $f^{-1} \sqbrk Y := \set {s \in S: \exists t \in Y: \map f s = t}$
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.
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 use counter image or inverse image instead of preimage.
Some sources hyphenate: pre-image.
Some sources use the notation $\map {f^\gets} Y$ for $f^{-1} \sqbrk Y$.
Also see
- Definition:Domain (Relation Theory)
- Definition:Codomain (Relation Theory)
- Definition:Range of Relation
- Results about preimages under mappings can be found here.
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
Sources
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next) $\S 2$