Definition:Image (Set Theory)/Mapping/Mapping

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Definition

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

Definition 1

The image of a mapping $f: S \to T$ is the set:

$\Img f = \set {t \in T: \exists s \in S: \map f s = t}$

That is, it is the set of values taken by $f$.

Definition 2

The image of a mapping $f: S \to T$ is the set:

$\Img f = f \sqbrk S$

where $f \sqbrk S$ is the image of $S$ under $f$.

Also denoted as

The notation $\Img f$ to denote the image of a mapping $f$ is specific to $\mathsf{Pr} \infty \mathsf{fWiki}$.

The usual notation is $\image f$ or a variant, but this is too easily confused with $\map \Im z$, the imaginary part of a complex number.

Hence the non-standard usage $\Img f$.

Also known as

Some sources refer to this as the direct image of a mapping, in order to differentiate it from an inverse image.

Rather than apply a mapping $f$ directly to a subset $A$, those sources often prefer to define the direct image mapping of $f$ as a separate concept in its own right.

Also seen is the term image set of mapping for $\Img f$.

Examples

Image of $\map f x = x^4 - 1$

Let $f: \R \to \R$ be the mapping defined as:

$\forall x \in \R: \map f x = x^4 - 1$

The image of $f$ is the unbounded closed interval:

$\Img f = \hointr {-1} \to$

and so $f$ is not a surjection.

Image of $\map f x = x^2 - 4 x + 5$

Let $f: \R \to \R$ be the mapping defined as:

$\forall x \in \R: \map f x = x^2 - 4 x + 5$

The image of $f$ is the unbounded closed interval:

$\Img f = \hointr 1 \to$

and so $f$ is not a surjection.