Antisymmetric relation

A binary relation R is antisymmetric if, for all elements a and b, aRb and bRa together imply a = b. Distinct elements may be related in one direction, but not both.

Binary relation: A relation between elements of two sets, represented as a set of ordered pairs. Antisymmetry is a condition imposed on a binary relation.
Partial order: A reflexive, antisymmetric, transitive relation on a set. Antisymmetry is one of the three defining conditions of a partial order.
Asymmetric relation: A relation is asymmetric when aRb always rules out bRa. Asymmetry forbids even self-relations, while antisymmetry allows them.
Logical implication: A logical connective stating that one proposition entails another. Antisymmetry is expressed as an implication from mutual relatedness to equality.
Ordered pair: A pair of objects in which the first and second positions are distinguished. Each instance of a binary relation is represented by an ordered pair.
Total order: A partial order in which every pair of elements is comparable. Total orders add comparability to the antisymmetric partial-order conditions.
Antireflexive relation: A relation is antireflexive when no element is related to itself. Antisymmetry places no restriction on self-relations; antireflexivity forbids all of them.
Directed graph: A graph whose edges have orientations from one vertex to another. A relation can be drawn as directed edges; antisymmetry rules out two-way edges between distinct vertices.
Reflexive relation: A relation on a set is reflexive when every element is related to itself. Reflexivity can hold alongside antisymmetry, as it does in partial orders.
Subset relation: The relation in which one set is related to another when it is a subset of it. If two sets contain each other, antisymmetry ensures they are equal.

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