Definition:Inverse Image Mapping/Mapping

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Definition

Let $S$ and $T$ be sets.

Let $\powerset S$ and $\powerset T$ be their power sets.

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


Definition 1

The inverse image mapping of $f$ is the mapping $f^\gets: \powerset T \to \powerset S$ that sends a subset $Y \subseteq T$ to its preimage $f^{-1} \paren T$ under $f$:

$\forall Y \in \powerset T: \map {f^\gets} Y = \begin {cases} \set {s \in S: \exists t \in Y: \map f s = t} & : \Img f \cap Y \ne \O \\ \O & : \Img f \cap Y = \O \end {cases}$


Definition 2

The inverse image mapping of $f$ is the direct image mapping of the inverse $f^{-1}$ of $f$:

$f^\gets = \paren {f^{-1} }^\to: \powerset T \to \powerset S$:

That is:

$\forall Y \in \powerset T: \map {f^\gets} Y = \set {s \in S: \exists t \in Y: \map f s = t}$


Inverse Image Mapping as Set of Preimages of Subsets

The inverse image mapping of $f$ can be seen to be the set of preimages of all the subsets of the codomain of $f$.

$\forall Y \subseteq T: f^{-1} \sqbrk Y = \map {f^\gets} Y$


Both approaches to this concept are used in $\mathsf{Pr} \infty \mathsf{fWiki}$.


Also defined as

Many authors define this concept only when $f$ is itself a mapping.


Also known as

The inverse image mapping of $f$ is also known as the preimage mapping of $f$.

Some sources refer to this as the mapping induced (on the power set) by the inverse $f^{-1}$.


Also denoted as

The notation used here is found in 1975: T.S. Blyth: Set Theory and Abstract Algebra.

The inverse image mapping can also be denoted $\map {\operatorname {\overline {\mathcal P} } } f$; see the contravariant power set functor.


Also see

  • Results about inverse image mappings can be found here.


Generalizations


Related Concepts