Definition:Image of Class under Mapping
Definition
Let $V$ be a basic universe
Let $f: V \to V$ be a mapping.
Let $A$ be a class.
The image of $A$ (under $f$) is defined and denoted as:
- $f \sqbrk A := \set {y: \exists x \in A: \map f x = y}$
That is, $f \sqbrk A$ is the class of all $\map F x$ where $x \in A$.
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.
Warning
Let $x \in A$ be a set of sets
Then:
- $f \sqbrk x$ is not necessarily the same as $\map f x$
where:
- $f \sqbrk x$ denotes the image of $x$ under $f$ where $x$ is treated as a class
- $\map f x$ denotes the image of $x$ under $f$ where $x$ is treated as an element of the class $A$.
Thus:
- $\map f x$ is what you get by applying $f$ to $x$
- $f \sqbrk x$ is what you get by applying $f$ to each of the elements of $x$ (but not $x$ itself) and then gathering the results into a set.
Also see
- Results about images can be found here.
Sources
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $6$: Order Isomorphism and Transfinite Recursion: $\S 1$ A few preliminaries: Definition $1.1$