Recognizing Symmetric Relations — Which relations are symmetric?

The sibling and friend relations are symmetric examples, while the < and father-child relations are not symmetric.

Symmetric Relation

Reflexivity Antisymmetry

A relation is symmetric if for any two elements: if (a,b) is in the relation, then (b,a) is also in it.

In other words: if the connection holds in one direction, it must also hold in the other direction.

Examples of Symmetric Relations

The "x is sibling of y" relation is symmetric: if Anna is sibling of Béla, then Béla is also sibling of Anna.
The "x is friend of y" relation (ideally) is symmetric: if Anna is friend of Béla, then Béla is also friend of Anna.

Counterexamples (Non-Symmetric Relations)

The "<" relation is not symmetric: if 2 < 5, it is not true that 5 < 2.
The "father-child" relation is also not symmetric: if Anna is father of Béla, it is still not true that Béla is father of Anna.

Summary

A relation is symmetric if it is always bidirectional. If we find a case where only one direction holds but not the other, then the relation is not symmetric.

Practice Exercise

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