# Definition:Image (Set Theory)/Mapping/Subset

## Contents

## Definition

Let $S$ and $T$ be sets.

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 and denoted as:

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

## Image of Subset as Element of Direct Image Mapping

The **image of $X$ under $f$** can be seen to be an element of the codomain of the direct image mapping $f^\to: \powerset S \to \powerset T$ of $f$:

- $\forall X \in \powerset S: \map {f^\to} X := \set {t \in T: \exists s \in X: \map f s = t}$

Thus:

- $\forall X \subseteq S: f \sqbrk X = \map {f^\to} X$

and so the **image of $X$ under $f$** is also seen referred to as the **direct image of $X$ under $f$**.

Both approaches to this concept are used in $\mathsf{Pr} \infty \mathsf{fWiki}$.

## Also known as

The term **image set** is often seen for **image**.

The modifier **by $f$** can also be used for **under $f$**.

Thus, for example, the **image set of $X$ by $f$** means the same as the **image of $X$ under $f$**.

## Notation

In parallel with the notation $f \sqbrk X$ for the direct image mapping of $f$, $\mathsf{Pr} \infty \mathsf{fWiki}$ also employs the notation $\map {f^\to} X$.

This latter notation is used in, for example, T.S. Blyth: *Set Theory and Abstract Algebra*, 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.

Some older sources use the notation $f \mathbin{``} X$ for $f \sqbrk X$.].

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.*- -- 1960: Paul R. Halmos:
*Naive Set Theory*

- -- 1960: Paul R. Halmos:

Similarly, Allan Clark: *Elements of Abstract Algebra*, which uses the notation $f x$ for what $\mathsf{Pr} \infty \mathsf{fWiki}$ denotes as $\map f x$, also uses $f X$ for $f \sqbrk X$ without comment on the implications.

In the same way does John D. Dixon: *Problems in Group Theory* 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 $\mathsf{Pr} \infty \mathsf{fWiki}$ 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.*- -- 1970: B. Hartley and T.O. Hawkes:
*Rings, Modules and Linear Algebra*

- -- 1970: B. Hartley and T.O. Hawkes:

## Examples

### Aribtrary Mapping from $\set {0, 1, 2, 3, 4, 5}$ to $\set {0, 1, 2, 3}$

Let:

\(\displaystyle S\) | \(=\) | \(\displaystyle \set {0, 1, 2, 3, 4, 5}\) | |||||||||||

\(\displaystyle T\) | \(=\) | \(\displaystyle \set {0, 1, 2, 3}\) |

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

\(\displaystyle f \paren 0\) | \(=\) | \(\displaystyle 0\) | |||||||||||

\(\displaystyle f \paren 1\) | \(=\) | \(\displaystyle 0\) | |||||||||||

\(\displaystyle f \paren 2\) | \(=\) | \(\displaystyle 0\) | |||||||||||

\(\displaystyle f \paren 3\) | \(=\) | \(\displaystyle 1\) | |||||||||||

\(\displaystyle f \paren 4\) | \(=\) | \(\displaystyle 1\) | |||||||||||

\(\displaystyle f \paren 5\) | \(=\) | \(\displaystyle 3\) |

Let:

\(\displaystyle A\) | \(=\) | \(\displaystyle \set {0, 3}\) | |||||||||||

\(\displaystyle B\) | \(=\) | \(\displaystyle \set {0, 1, 3}\) | |||||||||||

\(\displaystyle C\) | \(=\) | \(\displaystyle \set {0, 1, 2}\) |

Then:

\(\displaystyle f \sqbrk A\) | \(=\) | \(\displaystyle \set {0, 1}\) | |||||||||||

\(\displaystyle f \sqbrk B\) | \(=\) | \(\displaystyle \set {0, 1}\) | |||||||||||

\(\displaystyle f \sqbrk C\) | \(=\) | \(\displaystyle \set 0\) |

and:

- $\Img f = \set {0, 1, 3}$

### Image of $\closedint {-3} 2$ under $x \mapsto x^4 - 1$

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

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

The image of the closed interval $\closedint {-3} 2$ is:

- $f \sqbrk {\paren {\closedint {-3} 2} } = \closedint {-1} {80}$

## Also see

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

### Generalizations

### Related Concepts

## Sources

- 1955: John L. Kelley:
*General Topology*... (previous) ... (next): Chapter $0$: Functions - 1960: Paul R. Halmos:
*Naive Set Theory*... (previous) ... (next): $\S 8$: Functions - 1964: Iain T. Adamson:
*Introduction to Field Theory*... (previous) ... (next): $\S 1.3$ - 1964: Murray R. Spiegel:
*Theory and Problems of Complex Variables*... (previous) ... (next): $2$: Transformations - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 12$ - 1967: John D. Dixon:
*Problems in Group Theory*... (previous) ... (next): Introduction: Notation - 1967: George McCarty:
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*Introductory Real Analysis*... (previous) ... (next): $\S 1.3$: Functions and mappings. Images and preimages - 1970: B. Hartley and T.O. Hawkes:
*Rings, Modules and Linear Algebra*... (previous) ... (next): $\S 2.2$: Homomorphisms - 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $1$: Mappings: $\S 11$ - 1971: Robert H. Kasriel:
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