Category:Preimages under Relations

From ProofWiki
Jump to: navigation, search

This category contains results about Preimages under Relations.


Let $\mathcal R \subseteq S \times T$ be a relation.

Let $\mathcal R^{-1} \subseteq T \times S$ be the inverse relation to $\mathcal R$, defined as:

$\mathcal R^{-1} = \left\{{\left({t, s}\right): \left({s, t}\right) \in \mathcal R}\right\}$


Preimage of Element

Every $s \in S$ such that $\left({s, t}\right) \in \mathcal R$ 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:

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


This can also be written:

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


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


Preimage of Subset

Let $Y \subseteq T$.

The preimage of $Y$ under $\mathcal R$ is defined as:

$\mathcal R^{-1} \left [{Y}\right] := \left\{{s \in S: \exists y \in Y: \left({s, y}\right) \in \mathcal R}\right\}$


That is, the preimage of $Y$ under $\mathcal R$ is the image of $Y$ under $\mathcal R^{-1}$.


If no element of $Y$ has a preimage, then $\mathcal R^{-1} \left [{Y}\right] = \varnothing$.


Preimage of Relation

The preimage of $\mathcal R \subseteq S \times T$ is:

$\Preimg {\mathcal R} := \mathcal R^{-1} \left [{T}\right] = \set {s \in S: \exists t \in T: \tuple {s, t} \in \mathcal R}$