Math 2.0

Mathematical logic and foundations Definition

Quotient set

Builds on Equivalence classFunction

Definition

Let ∼\sim be an equivalence relation on a set AA. The quotient set of AA by ∼\sim is the set A/∼={[a]∣a∈A}A/{\sim} = \{[a] \mid a \in A\} of all equivalence classes. The canonical projection is the function π ⁣:A→A/∼\pi \colon A \to A/{\sim} with π(a)=[a]\pi(a) = [a].

Remarks

For congruence modulo nn on ℤ\mathbb{Z}, the quotient set ℤ/nℤ\mathbb{Z}/n\mathbb{Z} has the nn elements [0],[1],…,[n−1][0], [1], \dots, [n-1]. The canonical projection is surjective. A function on A/∼A/{\sim} is often defined by a formula on representatives. The formula must give the same value for equivalent representatives.

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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 pairdefinitionSet-builder notation, concept, no evidence levelSet-builder notationconceptExistential 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 relationdefinitionSymmetric relation, definition, no evidence levelSymmetric relationdefinitionTransitive relation, definition, no evidence levelTransitive relationdefinitionFunction, definition, no evidence levelFunctiondefinitionEquivalence relation, definition, no evidence levelEquivalence relationdefinitionEquivalence class, definition, no evidence levelEquivalence classdefinitionQuotient set, definition, no evidence levelQuotient setdefinition
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.
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  1. Element definition
  2. Proposition concept
  3. Set definition
  4. Implication definition
  5. Predicate definition
  6. Ordered pair definition
  7. Set-builder notation concept
  8. Existential quantifier definition
  9. Universal quantifier definition
  10. Cartesian product definition
  11. Subset definition
  12. Uniqueness quantifier definition
  13. Binary relation definition
  14. Reflexive relation definition
  15. Symmetric relation definition
  16. Transitive relation definition
  17. Function definition
  18. Equivalence relation definition
  19. Equivalence class definition
  20. Quotient set definition

Where this leads

24 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Equivalence_class, Definition and notation
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MSC03E20
DefinesQuotient set
UsesEquivalence class, Function

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