Math 2.0

Mathematical logic and foundations Definition

Function composition

Builds on Function

Definition

Let f ⁣:A→Bf \colon A \to B and g ⁣:B→Cg \colon B \to C be functions. The composition of gg and ff is the function g∘f ⁣:A→Cg \circ f \colon A \to C with (g∘f)(a)=g(f(a))(g \circ f)(a) = g(f(a)) for each a∈Aa \in A.

Remarks

Read g∘fg \circ f as "gg after ff": ff acts first. In general g∘f≠f∘gg \circ f \neq f \circ g. For f(x)=x+1f(x) = x + 1 and g(x)=2xg(x) = 2x on ℝ\mathbb{R}, (g∘f)(x)=2x+2(g \circ f)(x) = 2x + 2 and (f∘g)(x)=2x+1(f \circ g)(x) = 2x + 1. Composition is associative. See associativity of composition.

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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 compositiondefinition
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  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

Where this leads

30 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Function_composition, lead section
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