Mapping/Composition/Section

Definition

Let ${}L,\,M$ and ${}N$ denote sets, let

$F\colon L\longrightarrow M,x\longmapsto F(x),$

and

$G\colon M\longrightarrow N,y\longmapsto G(y),$

be mappings. Then the mapping

$G\circ F\colon L\longrightarrow N,x\longmapsto G(F(x)),$

is called the composition of the mappings

${}F$ and ${}G$.

So we have

${}(G\circ F)(x):=G(F(x))\,,$

where the left-hand side is defined by the right-hand side. If both mappings are given by functional expressions, then the composition is realized by plugging in the first term into the variable of the second term (and to simplify the expression, if possible).

For a bijective mapping ${}\varphi \colon M\rightarrow N$, the inverse mapping ${}\varphi ^{-1}\colon N\rightarrow M$ is characterized by the conditions

${}\varphi \circ \varphi ^{-1}=\operatorname {Id} _{N}\,$

and

${}\varphi ^{-1}\circ \varphi =\operatorname {Id} _{M}\,.$

Lemma

Let ${}L,M,N$ and ${}P$ be sets, and let

$F\colon L\longrightarrow M,x\longmapsto F(x),$

$G\colon M\longrightarrow N,y\longmapsto G(y),$

and

$H\colon N\longrightarrow P,z\longmapsto H(z),$

be mappings. Then

${}H\circ (G\circ F)=(H\circ G)\circ F\,$

holds.

Proof

Two mappings ${}\alpha ,\beta \colon L\rightarrow P$ are the same if and only if the equality ${}\alpha (x)=\beta (x)$ holds for every ${}x\in L$. So let ${}x\in L$. Then

${}{\begin{aligned}(H\circ (G\circ F))(x)&=H((G\circ F)(x))\\&=H(G(F(x)))\\&=(H\circ G)(F(x))\\&=((H\circ G)\circ F)(x).\end{aligned}}$ 

$\Box$