Symmetric relation explained

A symmetric relation is a type of binary relation. Formally, a binary relation R over a set X is symmetric if:

\forall a, b \in X(a R b \Leftrightarrow b R a) , where the notation aRb means that .

An example is the relation "is equal to", because if is true then is also true. If R^T represents the converse of R, then R is symmetric if and only if .^[1]

Symmetry, along with reflexivity and transitivity, are the three defining properties of an equivalence relation.

Examples

In mathematics

• "is equal to" (equality) (whereas "is less than" is not symmetric)
• "is comparable to", for elements of a partially ordered set
• "... and ... are odd":

Outside mathematics

• "is married to" (in most legal systems)
• "is a fully biological sibling of"
• "is a homophone of"
• "is a co-worker of"
• "is a teammate of"

Relationship to asymmetric and antisymmetric relations

By definition, a nonempty relation cannot be both symmetric and asymmetric (where if a is related to b, then b cannot be related to a (in the same way)). However, a relation can be neither symmetric nor asymmetric, which is the case for "is less than or equal to" and "preys on").

Symmetric and antisymmetric (where the only way a can be related to b and b be related to a is if) are actually independent of each other, as these examples show.

Mathematical examples

	Symmetric	Not symmetric
Antisymmetric		divides, less than or equal to
Not antisymmetric		// (integer division), most nontrivial permutations

Non-mathematical examples

	Symmetric	Not symmetric
Antisymmetric	is the same person as, and is married	is the plural of
Not antisymmetric	is a full biological sibling of	preys on

Properties

• A symmetric and transitive relation is always quasireflexive.
• One way to count the symmetric relations on n elements, that in their binary matrix representation the upper right triangle determines the relation fully, and it can be arbitrary given, thus there are as many symmetric relations as binary upper triangle matrices, 2^n(n+1)/2.

Notes and References

1. Web site: MAD3105 1.2 . Florida State University Department of Mathematics . Florida State University . 30 March 2024.