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

## Definition

Let $S$ and $T$ be sets.

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

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

### Definition 1

The **image of $X$ (under $f$)** is defined and denoted as:

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

### Definition 2

The **image of $X$ under $f$** is the 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 $f$**.

## Class-Theoretical Definition

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.

Let $C \subseteq A$.

The **image of $C$ under $f$** is defined as:

\(\ds f \sqbrk C\) | \(=\) | \(\ds \set {y \in B: \exists x \in C: \map f x = y}\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \set {\map f x: x \in C}\) |

## Also known as

The term **image set** is often seen for the **image of a subset under a mapping**.

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 $f \mathbin{``} X$ or $\map {f*} X$ for $f \sqbrk X$.*

Sources which use the notation $s f$ for $\map f s$ may also use $S f$ or $S^f$ for $f \sqbrk S$.

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:

The above discussion applies equally well to classes as to sets.

## Examples

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

Let:

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

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

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

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

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

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

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

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

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

Let:

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

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

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

Then:

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

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

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

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:

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

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

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

## 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

- 1981: Murray R. Spiegel:
*Theory and Problems of Complex Variables*(SI ed.) ... (previous) ... (next): $2$: Functions, Limits and Continuity: Transformations - 2002: Thomas Jech:
*Set Theory*(3rd ed.) ... (previous) ... (next): Chapter $1$: Power Set