# 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