Definition:Image of Subset under Mapping/Definition 1
Definition
Let $S$ and $T$ be sets.
Let $f: S \to T$ be a mapping.
Let $X \subseteq S$ be a subset of $S$.
The image of $X$ (under $f$) is defined and denoted as:
- $f \sqbrk X := \set {t \in T: \exists s \in X: \map f s = t}$
Also known as
The term image set is often seen for the image of a subset under a mapping.
The modifier by $f$ can also be used for under $f$.
Thus, for example, the image set of $X$ by $f$ means the same as the image of $X$ under $f$.
Notation
In parallel with the notation $f \sqbrk X$ for the direct image mapping of $f$, $\mathsf{Pr} \infty \mathsf{fWiki}$ also employs the notation $\map {f^\to} X$.
This latter notation is used in, for example, T.S. Blyth: Set Theory and Abstract Algebra, and is referred to as the mapping induced by $f$:
- It should be noted that most mathematicians write $\map f X$ for $\map {f^\to} X$. Now it is quite clear that the mappings $f$ and $f^\to$ are not the same, so we shall retain the notation $f^\to$ to avoid confusion. ... We shall say that the mappings $f^\to$ and $f^\gets$ are the mappings which are induced on the power sets by the mapping $f$.
In a similar manner, the notation $f^{-1} \sqbrk X$, for the premage of a subset under a mapping, otherwise known as the inverse image mapping of $f$, also has the notation $\map {f^\gets} X$ used for it.
Some older sources use the notation $f \mathbin{``} X$ or $\map {f} X$ for $f \sqbrk X$.
Sources which use the notation $s f$ for $\map f s$ may also use $S f$ or $S^f$ for $f \sqbrk S$.
Some authors do not bother to make the distinction between the image of an element and the image set of a subset, and use the same notation for both:
- The notation is bad but not catastrophic. What is bad about it is that if $A$ happens to be both an element of $X$ and a subset of $X$ (an unlikely situation, but far from an impossible one), then the symbol $\map f A$ is ambiguous. Does it mean the value of $f$ at $A$ or does it mean the set of values of $f$ at the elements of $A$? Following normal mathematical custom, we shall use the bad notation, relying on context, and, on the rare occasions when it is necessary, adding verbal stipulations, to avoid confusion.
- -- 1960: Paul R. Halmos: Naive Set Theory
Similarly, Allan Clark: Elements of Abstract Algebra, which uses the notation $f x$ for what $\mathsf{Pr} \infty \mathsf{fWiki}$ denotes as $\map f x$, also uses $f X$ for $f \sqbrk X$ without comment on the implications.
In the same way does John D. Dixon: Problems in Group Theory provide us with $S^f$ for $f \sqbrk S$ as an alternative to $\map f S$, again making no notational distinction between the image of the subset and the image of the element.
On $\mathsf{Pr} \infty \mathsf{fWiki}$ this point of view is not endorsed.
Some authors recognise the confusion, and call attention to it, but don't actually do anything about it:
- In this way we obtain a map from the set $\powerset X$ of subsets of $X$ to $\powerset Y$; this map is still denoted by $f$, although strictly speaking it should be given a different name.
- -- 1970: B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra
The above discussion applies equally well to classes as to sets.
Sources
- 1951: Nathan Jacobson: Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts ... (previous) ... (next): Introduction $\S 2$: Product sets, mappings
- 1955: John L. Kelley: General Topology ... (previous) ... (next): Chapter $0$: Functions
- 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): $\S 8$: Functions
- 1964: Iain T. Adamson: Introduction to Field Theory ... (previous) ... (next): Chapter $\text {I}$: Elementary Definitions: $\S 3$. Homomorphisms
- 1965: Claude Berge and A. Ghouila-Houri: Programming, Games and Transportation Networks ... (previous) ... (next): $1$. Preliminary ideas; sets, vector spaces: $1.1$. Sets
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 12$: Homomorphisms
- 1968: A.N. Kolmogorov and S.V. Fomin: Introductory Real Analysis ... (previous) ... (next): $\S 1.3$: Functions and mappings. Images and preimages
- 1970: B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra ... (previous) ... (next): $\S 2.2$: Homomorphisms
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $1$: Mappings: $\S 11$
- 1971: Robert H. Kasriel: Undergraduate Topology ... (previous) ... (next): $\S 1.10$: Functions: Definition $10.6$
- 1974: Thomas W. Hungerford: Algebra ... (previous) ... (next): $\S 1.2$
- 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 6$: Functions
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): Notation and Terminology
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 21$: The image of a subset of the domain; surjections
- 1986: Geoffrey Grimmett and Dominic Welsh: Probability: An Introduction ... (previous) ... (next): $\S 2.1$: Probability mass functions: Footnote
- 1993: Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (2nd ed.) ... (previous) ... (next): $\S 1$: Naive Set Theory: $\S 1.6$: Functions
- 1994: Martin J. Osborne and Ariel Rubinstein: A Course in Game Theory ... (previous) ... (next): Chapter $1$ Introduction: $1.7$: Terminology and Notation
- 1996: Winfried Just and Martin Weese: Discovering Modern Set Theory. I: The Basics ... (previous) ... (next): Part $1$: Not Entirely Naive Set Theory: Chapter $1$: Pairs, Relations, and Functions
- 1996: H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability ... (previous) ... (next): Appendix $\text{A}.3$: Functions
- 1999: András Hajnal and Peter Hamburger: Set Theory ... (previous) ... (next): $1$. Notation, Conventions: $10$
- 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 2$: Functions
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $\S 2$
- 2008: Paul Halmos and Steven Givant: Introduction to Boolean Algebras ... (previous) ... (next): Appendix $\text{A}$: Set Theory: Images and Inverse Images
- 2011: Robert G. Bartle and Donald R. Sherbert: Introduction to Real Analysis (4th ed.) ... (previous) ... (next): $\S 1.1$: Sets and Functions