Symmetrical relation

Three symmetric relation, as directed graphs shown

The symmetry of a two-digit relation R on a set is given if x R y always implies y R x . It is called R then symmetrically .

Symmetry is one of the prerequisites for an equivalence relation .

Terms contradicting symmetry are antisymmetry and asymmetry .

Formal definition

If there is a set and a two-digit relation , then it is called symmetric if (using the infix notation ) applies: ${\displaystyle M}$ ${\displaystyle R\subseteq M\times M}$ ${\displaystyle M}$ ${\displaystyle R}$

${\displaystyle \forall x,y\in M:(xRy\Rightarrow yRx)}$

Examples

Equality of real numbers

Ordinary equality on the real numbers is symmetric because from follows . It is also an equivalence relation . ${\displaystyle =}$ ${\displaystyle x=y}$ ${\displaystyle y=x}$

The inequality relation on the real numbers is not an equivalence relation, but it is also symmetrical, because it follows . ${\displaystyle \neq }$ ${\displaystyle x\neq y}$ ${\displaystyle y\neq x}$

Similarity of triangles

If the triangle ABC is similar to the triangle DEF, the triangle DEF is similar to the triangle ABC. So the relation of the similarity of triangles is symmetrical. It is also an equivalence relation.

Congruence modulo m

An integer a means to the integer b congruent modulo m (with the integer m ≠ 0, called module) when both a as b in the division by m have the same remainder. For example, the number 11 is congruent to the number 18 modulo 7, since dividing these two numbers by 7 results in the remainder 4. This relation is symmetrical. It is also an equivalence relation.

Order of real numbers

The smaller relation on the real numbers is not symmetrical, because and cannot hold at the same time. ${\displaystyle <}$ ${\displaystyle x<y}$ ${\displaystyle y<x}$

Representation as a directed graph

Any relation R on a set M can be understood as a directed graph (see example above). The nodes of the graph are the elements of M . A directed edge (an arrow ) is drawn from node a to node b if and only if a R b holds. ${\displaystyle a\longrightarrow b}$

The symmetry of R can now be characterized in the graph as follows: Whenever there is an arrow between different nodes a and b of the graph, then there is an arrow at the same time . (A graph with this property is also called a symmetric graph .) ${\displaystyle a\longrightarrow b}$ ${\displaystyle b\longrightarrow a}$

Arrows automatically meet this criterion. ${\displaystyle a\longrightarrow a}$

properties

• With the help of the converse relation , the symmetry of a relation can be characterized by ${\displaystyle R^{-1}}$ ${\displaystyle R}$

  ${\displaystyle R=R^{-1}}$

• If the relation is symmetrical, then this also applies to the complementary relation . This is defined by ${\displaystyle R}$ ${\displaystyle R^{\rm {c}}}$

  ${\displaystyle xR^{\rm {c}}y:\Longleftrightarrow \neg xRy}$.

• If the relations and are symmetric, then this also applies to their intersection and their union . This statement can be generalized from two relations to the intersection and the union of any (non-empty) family of symmetric relations. This forms a topological space with the symmetrical relations as open sets. Furthermore, the set of symmetric relations is then also a set algebra over . ${\displaystyle R}$ ${\displaystyle S}$ ${\displaystyle R\cap S}$ ${\displaystyle R\cup S}$ ${\displaystyle \cap _{i\in I}R_{i}}$ ${\displaystyle \cup _{i\in I}R_{i}}$ ${\displaystyle M\times M}$ ${\displaystyle M\times M}$
• The smallest symmetric relation that includes a given relation is called the symmetric closure of . This can easily be specified as ${\displaystyle S}$ ${\displaystyle R}$ ${\displaystyle R}$

  ${\displaystyle S:=R\cup R^{-1}}$

• For any two-digit relation on a set, the powers of the concatenation of relations can be formed. If it is symmetrical, then this also applies to all powers . ${\displaystyle R}$ ${\displaystyle R^{n}}$ ${\displaystyle R}$ ${\displaystyle R^{n}}$
• A relation (on a finite set) is symmetric if and only if the adjacency matrix assigned to its graph is symmetric (to the main diagonal ).