Math 2.0

Order, lattices, ordered algebraic structures Definition

Upper bound

Builds on Partial order

Definition

Let (P,≤)(P, \le) be a partially ordered set and let SS be a subset of PP. An element u∈Pu \in P is an upper bound of SS when s≤us \le u for each s∈Ss \in S. The set SS is bounded above when it has an upper bound.

Remarks

An upper bound need not be in SS. The number 11 is an upper bound of [0,1)[0, 1) in ℝ\mathbb{R}, and so is 55. A lower bound is defined in the same way with u≤su \le s. The least upper bound of SS, when it exists, is the supremum of SS.

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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 pairdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionAntisymmetric relation, definition, no evidence levelAntisymmetric relationdefinitionReflexive relation, definition, no evidence levelReflexive relationdefinitionTransitive relation, definition, no evidence levelTransitive relationdefinitionPreorder, definition, no evidence levelPreorderdefinitionPartial order, definition, no evidence levelPartial orderdefinitionUpper bound, definition, no evidence levelUpper bounddefinition
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  1. Element definition
  2. Proposition concept
  3. Set definition
  4. Implication definition
  5. Predicate definition
  6. Ordered pair definition
  7. Universal quantifier definition
  8. Cartesian product definition
  9. Subset definition
  10. Binary relation definition
  11. Antisymmetric relation definition
  12. Reflexive relation definition
  13. Transitive relation definition
  14. Preorder definition
  15. Partial order definition
  16. Upper bound definition

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14 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Upper_and_lower_bounds, lead section
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MSC06A06
DefinesUpper bound
UsesPartial order

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