Definition:Preimage/Relation/Element

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Definition

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:

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


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 $\mathcal R^{-1}$.


Warning

Note that:

  • $t \in T$ may have more than one preimage.
  • It is possible for $t \in T$ to have no preimages at all, in which case $\mathcal R^{-1} \left ({t}\right) = \varnothing$.


Also known as

The preimage of $t \in T$ is also known as:

the fiber of $t$
the preimage set of $t$
the inverse image of $t$.


As well as using the notation $\operatorname{Im}^{-1} \left ({\mathcal R}\right)$ to denote the preimage of an entire relation, the symbol $\operatorname{Im}^{-1}$ can also be used as follows:

For $t \in \operatorname{Im} \left({\mathcal R}\right)$:

$\operatorname{Im}^{-1}_\mathcal R \left ({t}\right) = \mathcal R^{-1} \left ({t}\right)$

but this notation is rarely seen.