Recognizing Antisymmetric Relations — Which relations are antisymmetric on natural numbers?

The ≤ and divisibility relations are antisymmetric, because if two elements are mutually related, they are identical. The < is also antisymmetric, because it cannot happen that both (a,b) and (b,a) are present. The sibling relation, however, is not antisymmetric.

Antisymmetric Relation

Symmetry Asymmetry

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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