Definition:Image of Subset under Mapping/Notation

Notation for Image of Subset under Mapping
In parallel with the notation $f \sqbrk X$ for the direct image mapping of $f$, also employs the notation $\map {f^\to} X$.

This latter notation is used in, for example,, 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.

As well as using the notation $\Img f$ to denote the image set of a mapping, the symbol $\operatorname {Img}$ can also be used as follows:


 * $\map {\operatorname {Img}_f} X := f \sqbrk X$

but this notation is rarely seen.

Similarly obscure is the notation $f \mathbin{``} X$ for $f \sqbrk X$, which is mainly encountered in older accounts on set theory.

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.

Similarly,, which uses the notation $f x$ for what denotes as $\map f x$, also uses $f X$ for $f \sqbrk X$ without comment on the implications.

In the same way does 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 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.