Math 2.0

Order, lattices, ordered algebraic structures Definition

Monotone function

Builds on PreorderFunction

Definition

Let (P,≤P)(P, \le_P) and (Q,≤Q)(Q, \le_Q) be preorders and let f ⁣:P→Qf \colon P \to Q be a function. The function ff is monotone when, for all a,b∈Pa, b \in P, a≤Pba \le_P b implies f(a)≤Qf(b)f(a) \le_Q f(b).

Remarks

Other names are order-preserving and, for real functions, monotonically increasing or non-decreasing. The function ff is antitone when a≤ba \le b implies f(b)≤f(a)f(b) \le f(a). The map S↦f(S)S \mapsto f(S) on power sets is monotone for ⊆\subseteq.

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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 relationdefinitionReflexive relation, definition, no evidence levelReflexive relationdefinitionTransitive relation, definition, no evidence levelTransitive relationdefinitionFunction, definition, no evidence levelFunctiondefinitionPreorder, definition, no evidence levelPreorderdefinitionMonotone function, definition, no evidence levelMonotone functiondefinition
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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. Reflexive relation definition
  14. Transitive relation definition
  15. Function definition
  16. Preorder definition
  17. Monotone function definition

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15 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Monotonic_function, In order theory
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Kinddefinition
WritingW1 stated
MSC06A06
DefinesMonotone function
UsesPreorder, Function

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