Math 2.0

Mathematical logic and foundations Definition

Equivalence class

Builds on Equivalence relationSet-builder notation

Definition

Let ∼\sim be an equivalence relation on a set AA and let a∈Aa \in A. The equivalence class of aa is the set [a]={x∈A∣x∼a}.[a] = \{x \in A \mid x \sim a\}. Each element of [a][a] is a representative of the class.

Remarks

For congruence modulo 33 on ℤ\mathbb{Z}, the class of 11 is [1]={…,−5,−2,1,4,7,… }[1] = \{\dots, -5, -2, 1, 4, 7, \dots\}. For a,b∈Aa, b \in A, [a]=[b][a] = [b] if and only if a∼ba \sim b. Otherwise, [a][a] and [b][b] are disjoint. The set of all classes is the quotient set.

Reading path

To understand this page from the ground up, read these 16 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 levelSetdefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionSet-builder notation, concept, no evidence levelSet-builder notationconceptUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionReflexive 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 classdefinition
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. Implication definition
  5. Predicate definition
  6. Ordered pair definition
  7. Set-builder notation concept
  8. Universal quantifier definition
  9. Cartesian product definition
  10. Subset definition
  11. Binary relation definition
  12. Reflexive relation definition
  13. Symmetric relation definition
  14. Transitive relation definition
  15. Equivalence relation definition
  16. Equivalence class definition

Where this leads

62 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Equivalence_class, lead section
About this page
Tag082RMRSY26
URIm2://082RMRSY26
Kinddefinition
WritingW1 stated
MSC03E20
DefinesEquivalence class
UsesEquivalence relation, Set-builder notation

Markdown · JSON · Edit on GitHub

What this page needs for W2

  • edge instance_of in: one example (0 of 1)
  • edge not_instance_of in: one non-example that fails one clause (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

Used by