Definition:Preimage/Mapping/Element
Definition
Let $f: S \to T$ be a mapping.
Let $f^{-1} \subseteq T \times S$ be the inverse of $f$, defined as:
- $f^{-1} = \set {\tuple {t, s}: \map f s = t}$
Every $s \in S$ such that $\map f s = t$ is called a preimage of $t$.
The preimage of an element $t \in T$ is defined as:
- $\map {f^{-1} } t := \set {s \in S: \map f s = t}$
This can also be expressed as:
- $\map {f^{-1} } t := \set {s \in \Img {f^{-1} }: \tuple {t, s} \in f^{-1} }$
That is, the preimage of $t$ under $f$ is the image of $t$ under $f^{-1}$.
Also known as
The preimage of an element is also known as its inverse image.
In other contexts, this is called the fiber of $t$ (under $f$).
The UK English spelling of fiber is fibre.
The term argument is popular in certain branches of mathematics.
If $\tuple {x, y} \in f$, then $x$ is the argument (of $f$) which holds the value $y$.
In the context of computability theory, the following terms are frequently found:
If $\tuple {x, y} \in f$, then $x$ is often called the input of $f$ which produces the output $y$.
Also see
- From Preimages All Exist iff Surjection, $\map {f^{-1} } t$ is guaranteed not to be empty if and only if $f$ is a surjection.
- From the definition of an injection, if $\map {f^{-1} } t$ is not empty, then it is guaranteed to be a singleton if and only if $f$ is an injection.
Thus, while $f^{-1}$ is always a relation, it is not actually a mapping unless $f$ is a bijection.
Sources
- 1951: Nathan Jacobson: Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts ... (previous) ... (next): Introduction $\S 3$: Equivalence relations
- 1959: E.M. Patterson: Topology (2nd ed.) ... (previous) ... (next): Chapter $\text {II}$: Topological Spaces: $\S 9$. Functions
- 1964: William K. Smith: Limits and Continuity ... (previous) ... (next): $\S 2.2$: Functions: Exercise $\text{A} \ 5 \ \text{(b)}$
- 1965: Claude Berge and A. Ghouila-Houri: Programming, Games and Transportation Networks ... (previous) ... (next): $1$. Preliminary ideas; sets, vector spaces: $1.1$. Sets
- 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 0.4$
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Functions
- 1968: A.N. Kolmogorov and S.V. Fomin: Introductory Real Analysis ... (previous) ... (next): $\S 1.3$: Functions and mappings. Images and preimages
- 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 2$: Sets and functions: Graphs and functions
- 1989: George S. Boolos and Richard C. Jeffrey: Computability and Logic (3rd ed.) ... (previous) ... (next): $1$ Enumerability
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): argument: 1.
- 1996: H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability ... (previous) ... (next): Appendix $\text{A}.3$: Functions
- 2008: Paul Halmos and Steven Givant: Introduction to Boolean Algebras ... (previous) ... (next): Appendix $\text{A}$: Set Theory: Functions
- 2008: David Joyner: Adventures in Group Theory (2nd ed.) ... (previous) ... (next): Chapter $2$: 'And you do addition?': $\S 2.1$: Functions