Math 2.0

Mathematical logic and foundations Theorem

Equivalence classes form a partition

Builds on Equivalence classPartition of a set

Statement

Let ∼\sim be an equivalence relation on a set AA. Then the set {[a]∣a∈A}\{[a] \mid a \in A\} of equivalence classes is a partition of AA.

Remarks

Each class [a][a] is nonempty, because a∈[a]a \in [a] by reflexivity. If [a][a] and [b][b] have a common element, then a∼ba \sim b by symmetry and transitivity, so [a]=[b][a] = [b]. The left cosets of a subgroup of a group form such a partition. This fact is the key step in Lagrange's theorem.

Reading path

To understand this page from the ground up, read these 27 pages in order. The foundations come first.

Show the path as a graph
Element, definition, no evidence levelElementdefinitionProposition, concept, no evidence levelPropositionconceptSet, definition, no evidence levelSetdefinitionConjunction, definition, no evidence levelConjunctiondefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionEmpty set, object, no evidence levelEmpty setobjectFamily of sets, definition, no evidence levelFamily of setsdefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionIntersection, definition, no evidence levelIntersectiondefinitionUnion, definition, no evidence levelUniondefinitionSet-builder notation, concept, no evidence levelSet-builder notationconceptExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionDisjoint sets, definition, no evidence levelDisjoint setsdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionUnion of a family of sets, definition, no evidence levelUnion of a family of …definitionBinary relation, definition, no evidence levelBinary relationdefinitionPartition of a set, definition, no evidence levelPartition of a setdefinitionReflexive relation, definition, no evidence levelReflexive relationdefinitionSymmetric relation, definition, no evidence levelSymmetric relationdefinitionTransitive relation, definition, no evidence levelTransitive relationdefinitionEquivalence relation, definition, no evidence levelEquivalence relationdefinitionEquivalence class, definition, no evidence levelEquivalence classdefinitionEquivalence classes form a partition, statement, no evidence levelEquivalence classes f…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. Conjunction definition
  5. Disjunction definition
  6. Implication definition
  7. Predicate definition
  8. Empty set object
  9. Family of sets definition
  10. Ordered pair definition
  11. Intersection definition
  12. Union definition
  13. Set-builder notation concept
  14. Existential quantifier definition
  15. Universal quantifier definition
  16. Disjoint sets definition
  17. Cartesian product definition
  18. Subset definition
  19. Union of a family of sets definition
  20. Binary relation definition
  21. Partition of a set definition
  22. Reflexive relation definition
  23. Symmetric relation definition
  24. Transitive relation definition
  25. Equivalence relation definition
  26. Equivalence class definition
  27. Equivalence classes form a partition statement

Where this leads

30 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Partition_of_a_set, Partitions and equivalence relations
About this page
Tag01HT2CH8BH
URIm2://01HT2CH8BH
Kindstatement, theorem
WritingW0 stub
MSC03E20
StatusThis copy of the site has no build data. Run m2 status 01HT2CH8BH.
UsesEquivalence class, Partition of a set

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