Definition:Image (Relation 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 $f$.


Class Theory

In the context of class theory, the definition follows the same lines:

Let $V$ be a basic universe.

Let $A \subseteq V$ and $B \subseteq V$ be classes.

Let $f: A \to B$ be a class mapping.

The image of $\RR$ is defined and denoted as:

$\Img \RR := \set {y \in V: \exists x \in V: \tuple {x, y} \in \RR}$

That is, it is the class of all $y$ such that $\tuple {x, y} \in \RR$ for at least one $x$.


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 $\map {\mathrm {Im} } f$ 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 $\Img f$.


Some sources use $f \sqbrk S$, where $S$ is the domain of $f$.

Others just use $\map f S$, but that notation is deprecated on $\mathsf{Pr} \infty \mathsf{fWiki}$ so as not to confuse it with the notation for the image of an element.


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.


In the context of set theory, the term image set for $\Img f$ can often be seen.


Examples

Arbitrary Example

Let $f$ be defined as:

$\forall x: 0 \le x \le 2: \map f x = x^3$

The image of $f$ is the closed interval $\closedint 0 8$.


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.


Also see

  • Results about images can be found here.


Sources

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