Definition:Image (Relation Theory)/Mapping/Subset

Definition
Let $f: S \to T$ be a mapping. Let $X \subseteq S$.

Then the image (or image set) of $X$ (by $f$) is defined as:
 * $f \left [{X}\right] := \left\{ {t \in T: \exists s \in X: f \left({s}\right) = t}\right\}$

That is:
 * $f \left [{X}\right] := f^\to \left ({X}\right)$

where $f^\to$ denotes the mapping induced on the power set of $S$ by $f$.

Notation
As well as using the notation $\operatorname{Im} \left ({f}\right)$ to denote the image set of a mapping, the symbol $\operatorname{Im}$ can also be used as follows:

For $X \subseteq S$, we have:
 * $\operatorname{Im}_f \left ({X}\right) := f \left [{X}\right]$

but this notation is rarely seen.

Similarly obscure is the notation $f\,`` X$ for $f \left [{X}\right]$.

Some authors prefer not to use the notation $f \left [{X}\right]$ and instead use the concept of the mapping induced from the power set of $S$ to the power set of $T$.

For example, uses $f^\to \left ({X}\right)$ for $f \left[{X}\right]$

Some authors stipulate the name further by calling $f \left [{X}\right]$ the direct image of $X$ (by $f$).

This is done to emphasize the distinction between this and the concept of the inverse image.

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 $f \left({A}\right)$ 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.

On this point of view is not endorsed.

Also see

 * Definition:Image of Subset under Relation


 * Image of Singleton under Mapping
 * Image of Domain of Mapping is Image Set
 * Image of Subset under Mapping equals Union of Images of Elements