Definition:Preimage/Mapping/Subset

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Definition

Let $f: S \to T$ be a mapping.

Let $f^{-1} \subseteq T \times S$ be the inverse of $f$, defined as:

$f^{-1} = \set {\tuple {t, s}: \map f s = t}$


Let $Y \subseteq T$.

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

$f^{-1} \sqbrk Y := \set {s \in S: \exists y \in Y: \map f s = y}$


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 $f^{-1} \sqbrk Y = \O$.


Also known as

Some sources use counter image or inverse image instead of preimage.


Also denoted as

When the language of induced mappings is used, then $\map {f^\gets} Y$ is seen for $f^{-1} \sqbrk Y$.


Also see


Generalization


Sources