Antisymmetric Relation
A relation is antisymmetric if for any two elements: if both (a,b) and (b,a) are in the relation, then it can only be that a = b.
In other words: two distinct elements cannot be in a mutual relationship. If both directions are present, then a and b are actually the same element.
Examples of Antisymmetric Relations
- The "≤" relation is antisymmetric: if a ≤ b and b ≤ a, then it can only be that a = b.
- The "divisibility" relation on natural numbers is antisymmetric: if a divides b and b divides a, then certainly a = b.
Counterexamples (Non-Antisymmetric Relations)
- The "sibling" relation is not antisymmetric: if Anna is a sibling of Béla and Béla is a sibling of Anna, they are still not the same person.
- The "friend of" relation is also not antisymmetric, because if Anna is a friend of Béla and Béla is a friend of Anna, they remain separate people.
Summary
A relation is antisymmetric if it cannot happen that two distinct elements are mutually related. If both directions are present, the elements must be identical.
Practice Exercise
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