Definition:Inverse of Mapping
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- Not to be confused with Definition:Inverse Mapping.
Definition
Let $f: S \to T$ be a mapping.
The inverse of $f$ is its inverse relation, defined as:
- $f^{-1} := \set {\tuple {t, s}: \map f s = t}$
That is:
- $f^{-1} := \set {\tuple {t, s}: \tuple {s, t} \in f}$
That is, $f^{-1} \subseteq T \times S$ is the relation which satisfies:
- $\forall s \in S: \forall t \in T: \tuple {t, s} \in f^{-1} \iff \tuple {s, t} \in f$
Also known as
The inverse of a mapping is also known as its converse.
Also denoted as
For the inverse of a mapping, some sources use the notation $f^\gets$ or $f^{\circ-1}$ instead of $f^{-1}$.
Also see
- Definition:Preimage of Mapping (also known as inverse image)
- Inverse of Mapping is One-to-Many Relation where it is demonstrated that $f^{-1}$ is in general not itself a mapping.
- Results about inverses of mappings can be found here.
Sources
- 1959: E.M. Patterson: Topology (2nd ed.) ... (previous) ... (next): Chapter $\text {II}$: Topological Spaces: $\S 9$. Functions
- 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 10$: Inverses and Composites
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 5$: Composites and Inverses of Functions
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Problem $\text{AA}$: Relations
- 1971: Robert H. Kasriel: Undergraduate Topology ... (previous) ... (next): $\S 1.11$: Relations