Inclusion illustration

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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 .


For quantities and with , the inclusion mapping is through the mapping rule

given. Sometimes the special arrow symbol is used for identification and then you write .

One speaks of a real inclusion if is a real subset of , that is, if there are elements in .


  • Each inclusion map is injective . True inclusion is not surjective .
  • If so, inclusion is the mapping of identity .
  • 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.
  • 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
  • Conversely, each inclusion mapping can be understood as a restriction of a suitable identical mapping :

Web links

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