Math 2.0

Mathematical logic and foundations Definition

Antisymmetric relation

Builds on Binary relation

Definition

Let RR be a binary relation on a set AA. The relation RR is antisymmetric when, for all a,b∈Aa, b \in A, aRba \mathrel{R} b and bRab \mathrel{R} a imply a=ba = b.

Remarks

The relation ≤\le on ℝ\mathbb{R} and the relation ⊆\subseteq on sets are antisymmetric. Divisibility is antisymmetric on ℕ\mathbb{N} but not on ℤ\mathbb{Z}, because 1∣−11 \mid -1 and −1∣1-1 \mid 1. Antisymmetric is not the opposite of symmetric: equality is both symmetric and antisymmetric.

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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 relationdefinition
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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

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Further reading

  1. Web https://en.wikipedia.org/wiki/Antisymmetric_relation, lead section
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