Symmetric relation

The symmetry of a binary relation R on a set is given, if for x R y implies y R x follows. We call R then symmetrical.

Symmetry is one of the prerequisites for an equivalence relation.

For symmetry contrasting terms are anti-symmetry and asymmetry.

Formal definition

Is a set and a binary relation on, then is called symmetric if and only if (using infix notation ):

Examples

Equality of real numbers

The usual equality on the reals is symmetric, because follows. She is also an equivalence relation.

The inequality relation on the real numbers is indeed no equivalence relation, but also symmetric, as follows.

Similarity of triangles

If the triangle ABC similar to the triangle DEF, the triangle DEF to triangle ABC is similar. The relation of similarity of triangles is therefore symmetrical. She is also an equivalence relation.

Congruence modulo n

A natural number a is said to be the natural number b congruent modulo n if a and b when divided by n the same remainder. For example, the number 11 to the number 18 congruent modulo 7 because each of the remaining 4 results in the division of these two numbers by 7. This relation is symmetric. She is also an equivalence relation.

Order of the real numbers

The less-than relation on the real numbers is not symmetric, because and can not be considered simultaneously.

Representation as a directed graph

Any relation R on a set M can be regarded as a directed graph (see example above). The nodes of the graph are the elements of M. From the node a to node b is a directed edge if and only pulled (an arrow ) when a R b is true.

The symmetry of R can now characterize the graph like this: Whenever there is an arrow between nodes a and b of the graph, then there is an arrow at the same time. ( A graph with this property is also called symmetric graph. )

Arrows do not play a role in this criterion.

Properties

• Using the converse relation can the symmetry of a relation by characterizing
• The relation symmetrical, this also applies to the complementary relation. This is defined by.
• Are the relations and symmetrical, then this also applies to their intersection and their union. This statement can be generalized from two relations on the average and the union of any ( non-empty ) family of symmetric relations.
• The smallest symmetric relation that includes a given relation, the symmetric completion is called from. This is easy to specify as
• For any binary relation on a set can be the powers concerning the concatenation of relations form. Is now symmetric, then this also applies to all powers.
• A relation (on a finite set ) is symmetric if its graph associated adjacency matrix is symmetric ( the main diagonal).

• Set theory
• Mathematical concept