Definition:Preimage of Subset under Mapping/Definition 2

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 $f$ can be seen to be an element of the codomain of the inverse image mapping $f^\gets: \powerset T \to \powerset S$ of $f$:

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

Thus:

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

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.

Some sources use the notation $\map {f^\gets} Y$ for $f^{-1} \sqbrk Y$.

Also see

• Equivalence of Definitions of Preimage of Subset under Mapping

Sources

• 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Functions
• 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 5$. Induced mappings; composition; injections; surjections; bijections