Definition:Composition of Mappings

From ProofWiki
Jump to navigation Jump to search

Definition

Let $f_1: S_1 \to S_2$ and $f_2: S_2 \to S_3$ be mappings such that the domain of $f_2$ is the same set as the codomain of $f_1$.


Definition 1

The composite mapping $f_2 \circ f_1$ is defined as:

$\forall x \in S_1: \map {\paren {f_2 \circ f_1} } x := \map {f_2} {\map {f_1} x}$


Definition 2

The composite of $f_1$ and $f_2$ is defined and denoted as:

$f_2 \circ f_1 := \set {\tuple {x, z} \in S_1 \times S_3: \tuple {\map {f_1} x, z} \in f_2}$


Definition 3

The composite of $f_1$ and $f_2$ is defined and denoted as:

$f_2 \circ f_1 := \set {\tuple {x, z} \in S_1 \times S_3: \exists y \in S_2: \map {f_1} x = y \land \map {f_2} y = z}$


CompositeMapping.png

General Definition

Let $f_1: S_1 \to S_2, f_2: S_2 \to S_3, \ldots, f_n: S_n \to S_{n + 1}$ be mappings such that the domain of $f_k$ is the same set as the codomain of $f_{k - 1}$.


Then the composite of $f_1, f_2, \ldots, f_n$ is defined and denoted as:

\(\ds \forall x \in S_1: \, \) \(\ds \map {\paren {f_n \circ \cdots \circ f_2 \circ f_1} } x\) \(:=\) \(\ds \begin {cases} \map {f_1} x & : n = 1 \\ \map {f_n} {\map {\paren {f_{n - 1} \circ \cdots \circ f_2 \circ f_1} } x} : & n > 1 \end {cases}\)
\(\ds \) \(=\) \(\ds \map {f_n} {\dotsm \map {f_2} {\map {f_1} x} \dotsm}\)


Commutative Diagram

The concept of composition of mappings can be illustrated by means of a commutative diagram.

This diagram illustrates the specific example of $f_2 \circ f_1$:

$\begin{xy}\xymatrix@+1em{ S_1 \ar[r]^*+{f_1} \ar@{-->}[rd]_*[l]+{f_2 \mathop \circ f_1} & S_2 \ar[d]^*+{f_2} \\ & S_3 }\end{xy}$


Composition as a Binary Operation

Let $\sqbrk {S \to S}$ be the set of all mappings from a set $S$ to itself.

Then the concept of composite mapping defines a binary operation on $\sqbrk {S \to S}$:

$\forall f, g \in \sqbrk {S \to S}: g \circ f = \set {\tuple {s, t}: s \in S, \tuple {f \paren s, t} \in g} \in \sqbrk {S \to S}$


Thus, for every pair $\tuple {f, g}$ of mappings in $\sqbrk {S \to S}$, the composition $g \circ f$ is another element of $\sqbrk {S \to S}$.


Warning

Let $f_1: S_1 \to S_2$ and $f_2: S_2 \to S_3$ be mappings such that:

$\Dom {f_2} \ne \Cdm {f_1}$

where $\Dom {f_2}$ and $\Cdm {f_1}$ denote domain and codomain respectively.

Then the composite mapping $f_2 \circ f_1$ is not defined.


Compare with the definition of composition of relations in the context of the fact that a mapping is a special kind of relation.


Also known as

In the context of analysis, this is often found referred to as a function of a function, which (according to some sources) makes set theorists wince, as it is technically defined as a function on the codomain of a function.

Such a composition is also known as a composite mapping or composite function.

Some sources call $f_2 \circ f_1$ the resultant of $f_1$ and $f_2$ or the product of $f_1$ and $f_2$, but neither of these is endorsed on $\mathsf{Pr} \infty \mathsf{fWiki}$.


Some authors write $f_2 \circ f_1$ as $f_2 f_1$.

Some use the notation $f_2 \cdot f_1$ or $f_2 . f_1$.

Some use the notation $f_2 \bigcirc f_1$.

Others, particularly in books having ties with computer science, write $f_1; f_2$ or $f_1 f_2$ (note the reversal of order), which is read as (apply) $f_1$, then $f_2$.


Examples

Compositions of $x^2$ with $2 x + 1$

Let $f: \R \to \R$ be the real function defined as:

$\forall x \in \R: \map f x = x^2$

Let $g: \R \to \R$ be the real function defined as:

$\forall x \in \R: \map g x = 2 x + 1$


Then the compositions of $f$ with $g$ are:

$f \circ g: \R \to \R$:

$\forall x \in \R: \map {\paren {f \circ g} } x = \paren {2 x + 1}^2$

$g \circ f: \R \to \R$:

$\forall x \in \R: \map {\paren {g \circ f} } x = 2 x^2 + 1$


Compositions of $\sin x$ with $2 x + 1$

Let $f: \R \to \R$ be the real function defined as:

$\forall x \in \R: \map f x = \sin x$

Let $g: \R \to \R$ be the real function defined as:

$\forall x \in \R: \map g x = 2 x + 1$


Then the compositions of $f$ with $g$ are:

$f \circ g: \R \to \R$:

$\forall x \in \R: \map {\paren {f \circ g} } x = \map \sin {2 x + 1}$

$g \circ f: \R \to \R$:

$\forall x \in \R: \map {\paren {g \circ f} } x = 2 \sin x + 1$


Compositions of $x^2 + 1$ with $x + 1$

Let $f: \R \to \R$ be the real function defined as:

$\forall x \in \R: \map f x = x^2 + 1$

Let $g: \R \to \R$ be the real function defined as:

$\forall x \in \R: \map g x = x + 1$


Then the compositions of $f$ with $g$ are:

\(\ds f \circ g: \R \to \R: \, \) \(\ds \map {\paren {f \circ g} } x\) \(=\) \(\ds \paren {x + 1}^2 + 1\) \(\ds = x^2 + 2 x + 2\)
\(\ds g \circ f: \R \to \R: \, \) \(\ds \map {\paren {g \circ f} } x\) \(=\) \(\ds \paren {x^2 + 1} + 1\) \(\ds = x^2 + 2\)


Arbitrary Finite Sets

Let:

\(\ds A\) \(=\) \(\ds \set {1, 2, 3}\)
\(\ds B\) \(=\) \(\ds \set {a, b}\)
\(\ds C\) \(=\) \(\ds \set {u, v, w}\)


Let $\theta: A \to B$ and $\phi: B \to C$ be defined in two-row notation as:

\(\ds \theta\) \(=\) \(\ds \binom {1 \ 2 \ 3} {a \ b \ a}\)
\(\ds \phi\) \(=\) \(\ds \binom {a \ b} {w \ v}\)


Then:

$\phi \circ \theta = \dbinom {1 \ 2 \ 3} {w \ v \ w}$


Arbitrary Finite Set with Itself

Let $X = Y = \set {a, b}$.


Consider the mappings from $X$ to $Y$:

\(\text {(1)}: \quad\) \(\ds \map {f_1} a\) \(=\) \(\ds a\)
\(\ds \map {f_1} b\) \(=\) \(\ds b\)


\(\text {(2)}: \quad\) \(\ds \map {f_2} a\) \(=\) \(\ds a\)
\(\ds \map {f_2} b\) \(=\) \(\ds a\)


\(\text {(3)}: \quad\) \(\ds \map {f_3} a\) \(=\) \(\ds b\)
\(\ds \map {f_3} b\) \(=\) \(\ds b\)


\(\text {(4)}: \quad\) \(\ds \map {f_4} a\) \(=\) \(\ds b\)
\(\ds \map {f_4} b\) \(=\) \(\ds a\)


The Cayley table illustrating the compositions of these $4$ mappings is as follows:

$\begin{array}{c|cccc} \circ & f_1 & f_2 & f_3 & f_4 \\ \hline f_1 & f_1 & f_2 & f_3 & f_4 \\ f_2 & f_2 & f_2 & f_2 & f_2 \\ f_3 & f_3 & f_3 & f_3 & f_3 \\ f_4 & f_4 & f_3 & f_2 & f_1 \\ \end{array}$

We have that $f_1$ is the identity mapping and is also the identity element in the algebraic structure under discussion.


Also see

  • Results about composite mappings can be found here.


Sources


This page may be the result of a refactoring operation.
As such, the following source works, along with any process flow, will need to be reviewed.
When this has been completed, the citation of that source work (if it is appropriate that it stay on this page) is to be placed above this message, into the usual chronological ordering.
In particular: Any of three (or more) definitions
If you have access to any of these works, then you are invited to review this list, and make any necessary corrections.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{SourceReview}} from the code.


Retrieved from "https://proofwiki.org/w/index.php?title=Definition:Composition_of_Mappings&oldid=659323"