Recognizing Asymmetric Relations — Which relations are asymmetric on natural numbers?

Q: Recognizing Asymmetric Relations — Which relations are asymmetric on natural numbers?

The < and strict divisor relations are asymmetric, because the truth of one direction excludes the other. The ≤ is not asymmetric, because it can include mutual cases (e.g., a = b). The sibling relation is also not asymmetric, because it can hold mutually.

Asymmetric Relation

Antisymmetry Transitivity

A relation is asymmetric if for any two elements: if (a,b) is in the relation, then (b,a) is definitely not in it.

In other words: if the connection is true in one direction, it can never be true in the other direction.

Examples of Asymmetric Relations

The "<" relation is asymmetric: if 2 < 5, then it is never true that 5 < 2.
The "strict divisor" relation is also asymmetric: if 2 is a strict divisor of 4, then 4 can never be a strict divisor of 2.

Counterexamples (Non-Asymmetric Relations)

The "≤" relation is not asymmetric, because if a = b, then both (a,b) and (b,a) are in it.
The "friend of" relation is also not asymmetric, because it is often mutual.

Summary

An asymmetric relation always excludes mutuality: if (a,b) is true, then (b,a) is definitely false. This is stricter than the antisymmetric condition, because self-relations (a,a) are allowed there, but not here.

Practice Exercise

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