Definition:Preimage/Relation
Definition
Let $\RR \subseteq S \times T$ be a relation.
Let $\RR^{-1} \subseteq T \times S$ be the inverse relation to $\RR$, defined as:
- $\RR^{-1} = \set {\tuple {t, s}: \tuple {s, t} \in \RR}$
Preimage of Element
Every $s \in S$ such that $\tuple {s, t} \in \RR$ 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:
- $\map {\RR^{-1} } t := \set {s \in S: \tuple {s, t} \in \RR}$
This can also be written:
- $\map {\RR^{-1} } t := \set {s \in \Img {\RR^{-1} }: \tuple {t, s} \in \RR^{-1} }$
That is, the preimage of $t$ under $\RR$ is the image of $t$ under $\RR^{-1}$.
Preimage of Subset
Let $Y \subseteq T$.
The preimage of $Y$ under $\RR$ is defined as:
- $\RR^{-1} \sqbrk Y := \set {s \in S: \exists t \in Y: \tuple {s, t} \in \RR}$
That is, the preimage of $Y$ under $\RR$ is the image of $Y$ under $\RR^{-1}$:
- $\RR^{-1} \sqbrk Y := \set {s \in S: \exists t \in Y: \tuple {t, s} \in \RR^{-1} }$
If no element of $Y$ has a preimage, then $\RR^{-1} \sqbrk Y = \O$.
Preimage of Relation
The preimage of $\RR \subseteq S \times T$ is:
- $\Preimg \RR := \RR^{-1} \sqbrk T = \set {s \in S: \exists t \in T: \tuple {s, t} \in \RR}$
Also known as
A preimage is also known as an inverse image.
Also see
- Definition:Domain (Relation Theory)
- Definition:Codomain (Relation Theory)
- Definition:Range of Relation
- Definition:Image of Relation
- Results about preimages under relations 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