Inclusion illustration

Two examples of inclusion. Ex b) shows real inclusion .

An inclusion mapping (short- inclusion ), natural embedding or canonical embedding is a mathematical function that a subset to its basic amount embeds .

definition

For quantities and with , the inclusion mapping is through the mapping rule ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle A\subseteq B}$ ${\displaystyle i\colon A\rightarrow B}$

${\displaystyle i(x)=x}$

given. Sometimes the special arrow symbol is used for identification and then you write . ${\displaystyle \hookrightarrow }$ ${\displaystyle i\colon A\hookrightarrow B}$

One speaks of a real inclusion if is a real subset of , that is, if there are elements in . ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle B\setminus A}$

properties

• Each inclusion map is injective . True inclusion is not surjective .
• If so, inclusion is the mapping of identity . ${\displaystyle A=B}$
• Any function can be broken down with regard to the concatenation of functions as , where is surjective and injective: Let be the image set of and be the function that corresponds to with , that is . For one take the inclusion picture. ${\displaystyle f\colon A\to B}$ ${\displaystyle f=h\circ g}$ ${\displaystyle g}$ ${\displaystyle h}$ ${\displaystyle C=\operatorname {im} f\subseteq B}$ ${\displaystyle f}$ ${\displaystyle g\colon A\to C}$ ${\displaystyle A}$ ${\displaystyle f}$ ${\displaystyle g(x)=f(x)}$ ${\displaystyle h\colon C\to B}$
• If there is an arbitrary function and a subset of the definition set , then one understands the restriction of to that function which corresponds to with . With the help of inclusion , the restriction can be briefly written as ${\displaystyle f\colon A\to B}$ ${\displaystyle X}$ ${\displaystyle A}$ ${\displaystyle f|_{X}}$ ${\displaystyle f}$ ${\displaystyle X}$ ${\displaystyle g\colon X\to B}$ ${\displaystyle X}$ ${\displaystyle f}$ ${\displaystyle i\colon X\to A}$

${\displaystyle f|_{X}=f\circ i}$.

• Conversely, each inclusion mapping can be understood as a restriction of a suitable identical mapping : ${\displaystyle i\colon A\hookrightarrow B}$ ${\displaystyle i=\left(\operatorname {id} _{B}\right)|_{A}}$

Web links

Wiktionary: Inclusion  - explanations of meanings, word origins, synonyms, translations

• Eric W. Weisstein : Inclusion Map . In: MathWorld (English).
• Koro: Inclusion mapping . In: PlanetMath . (English)