Definition:Image (Set Theory)

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Definition

The definition of a relation as a subset of the Cartesian product of two sets gives a "static" sort of feel to the concept.

However, we can also consider a relation as being an operator, where you feed an element $s \in S$ (or a subset $S_1 \subseteq S$) in at one end, and you get a set of elements $T_s \subseteq T$ out of the other.

Thus we arrive at the following definition.


Relation

Image of a Relation

The image of $\mathcal R$ is the set:

$\Img {\mathcal R} := \mathcal R \left [{S}\right] = \set {t \in T: \exists s \in S: \tuple {s, t} \in \mathcal R}$


Image of an Element

Let $s \in S$.

The image of $s$ by (or under) $\mathcal R$ is defined as:

$\mathcal R \paren s := \set {t \in T: \tuple {s, t} \in \mathcal R}$

That is, $\mathcal R \paren s$ is the set of all elements of the codomain of $\mathcal R$ related to $s$ by $\mathcal R$.


Image of a Subset

Let $X \subseteq S$ be a subset of $S$.


Then the image set (of $X$ by $\mathcal R$) is defined as:

$\mathcal R \left [{X}\right] := \set {t \in T: \exists s \in X: \left({s, t}\right) \in \mathcal R}$


Mapping

Image of a Mapping

Definition 1

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

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

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 $f$.


Image of an Element

Let $s \in S$.

The image of $s$ (under $f$) is defined as:

$\Img s = \map f s = \displaystyle \bigcup \set {t \in T: \tuple {s, t} \in f}$

That is, $\map f s$ is the element of the codomain of $f$ related to $s$ by $f$.


Image of a Subset

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

Let $X \subseteq S$ be a subset of $S$.

Then the image of $X$ (under $f$) is defined as:

$f \sqbrk X := \set {t \in T: \exists s \in X: f \paren s = t}$



Also known as

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

Rather than apply a relation $\mathcal R$ (or mapping $f$) directly to a subset $A$, those sources prefer to define the mapping induced by $\mathcal R$ or $f$ as a separate concept in its own right.


Also see


Technical Note

The $\LaTeX$ code for \(\Img {f}\) is \Img {f} .

When the argument is a single character, it is usual to omit the braces:

\Img f