Math 2.0

Mathematical logic and foundations Proposition

Associativity of composition

Builds on Function composition

Statement

Let f ⁣:A→Bf \colon A \to B, g ⁣:B→Cg \colon B \to C and h ⁣:C→Dh \colon C \to D be functions. Then h∘(g∘f)=(h∘g)∘f.h \circ (g \circ f) = (h \circ g) \circ f.

Remarks

Both sides send each a∈Aa \in A to h(g(f(a)))h(g(f(a))), so they are equal. Because of this law, one can write h∘g∘fh \circ g \circ f without brackets. With the identity functions, the functions from a set to itself form a monoid under composition.

Reading path

To understand this page from the ground up, read these 15 pages in order. The foundations come first.

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Element, definition, no evidence levelElementdefinitionProposition, concept, no evidence levelPropositionconceptSet, definition, no evidence levelSetdefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionFunction, definition, no evidence levelFunctiondefinitionFunction composition, definition, no evidence levelFunction compositiondefinitionAssociativity of composition, statement, no evidence levelAssociativity of comp…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Element definition
  2. Proposition concept
  3. Set definition
  4. Implication definition
  5. Predicate definition
  6. Ordered pair definition
  7. Existential quantifier definition
  8. Universal quantifier definition
  9. Cartesian product definition
  10. Subset definition
  11. Uniqueness quantifier definition
  12. Binary relation definition
  13. Function definition
  14. Function composition definition
  15. Associativity of composition statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Function_composition, Properties
About this page
Tag0J6AKYZQT8
URIm2://0J6AKYZQT8
Aliasescomposition-is-associative
Kindstatement, proposition
WritingW0 stub
MSC03E20
StatusThis copy of the site has no build data. Run m2 status 0J6AKYZQT8.
UsesFunction composition

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