Relation (Mathematik)

Binare relation reflexiv

The complement of a reflexive relation is irreflexive—and vice versa.

BM7. Binary Relations

The complement of a strict weak order is a total preorder—and vice versa. If a relation is reflexiveirreflexive, symmetricantisymmetricasymmetrictransitivetotaltrichotomousa partial ordertotal orderstrict weak ordertotal preorder weak orderor an equivalence binare relation reflexivthen so are its restrictions too.

However, the transitive closure of a restriction is a subset of the restriction of the transitive closure, i.

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For example, restricting the relation "x is parent of y" to females yields the relation "x is mother of the woman y"; its transitive closure doesn't relate a woman with her paternal grandmother. On the other hand, the transitive closure of "is parent of" is "is ancestor of"; onlineverdienstprogramm restriction to females does relate a woman with her paternal grandmother.

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Also, the various concepts of completeness not to be confused with being "total" do not carry over to restrictions. Matrix representation[ edit ] Binary relations over sets X and Y can be represented algebraically by logical matrices indexed by X and Y with entries in the Boolean semiring addition corresponds to OR and multiplication to AND where matrix addition corresponds to union of relations, matrix multiplication corresponds to composition of relations of a relation over X and Y and a relation over Y and Z[18] the Hadamard product corresponds to intersection of relations, the zero matrix corresponds to the empty relation, and the matrix of ones corresponds to the universal relation.

In most mathematical contexts, references to the relations of equality, membership and subset are harmless because they can be understood implicitly to be restricted to some set in the context.

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Another solution to this problem is to use a set theory with proper classes, such as NBG or Morse—Kelley set theoryand allow the domain and codomain and so the graph to be proper classes : in such a theory, equality, membership, and subset are binary relations without special comment.

A minor modification needs to be made to the concept of the ordered triple X, Y, Gas normally a proper class cannot be a member of an ordered tuple; or of course one can identify the binary relation with its graph in this context.

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Homogeneous relation[ edit ] A homogeneous relation also called endorelation over a set X is a binary relation over X and itself, i. An example of a homogeneous relation is the relation of kinshipwhere the relation is over people.

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A homogeneous relation R binare relation reflexiv a set X may be identified with a directed simple graph permitting loopsor if it is symmetricwith an undirected simple graph permitting loopswhere X is the vertex set and R is the edge set there is an edge from a vertex x to a vertex y if and only if xRy. It is called the adjacency relation of the graph. The set of all homogeneous relations B.

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