# Logic characteristics of binary relationships

This difference in perspectives does raise some nontrivial issues. As an example, the former camp considers surjectivity —or being onto—as a property of functions, while the latter sees it as a relationship that functions may bear to sets. Either approach is adequate for most uses, provided that one attends to the necessary changes in language, notation, and the definitions of concepts like restrictions , composition , inverse relation , and so on.

The choice between the two definitions usually matters only in very formal contexts, like category theory. Suppose that John owns the ball, Mary owns the doll, and Venus owns the car. Nobody owns the gun and Ian owns nothing. Then the binary relation "is owned by" is given as. Thus the first element of R is the set of objects, the second is the set of persons, and the last element is a set of ordered pairs of the form object, owner.

The pair ball, John , denoted by ball R John means that the ball is owned by John. But the graphs of the two relations are the same. Some important types of binary relations R between two sets X and Y are listed below. To emphasize that X and Y can be different sets, some authors call such binary relations heterogeneous. Totality properties only definable if the sets of departure X resp. As examples, any function or any functional right-unique relation is difunctional; the converse doesn't hold.

A characterization of difunctional relations, which also explains their name, is to consider two functions f: In automata theory , the term rectangular relation has also been used to denote a difunctional relation. This terminology is justified by the fact that when represented as a boolean matrix, the columns and rows of a difunctional relation can be arranged in such a way as to present rectangular blocks of true on the asymmetric main diagonal.

For the theoretical explanation see Relation algebra. A relation that is reflexive, symmetric, and transitive is called an equivalence relation. A relation that is symmetric, transitive, and serial is also reflexive. A relation that is only symmetric and transitive without necessarily being reflexive is called a partial equivalence relation.

A relation that is reflexive, antisymmetric, and transitive is called a partial order. A partial order that is total is called a total order , simple order , linear order, or a chain. In this case, if R and S disagree, R is also said to be smaller than S. If R is a binary relation over X , then each of the following is a binary relation over X:.

The restriction of a binary relation on a set X to a subset S is the set of all pairs x , y in the relation for which x and y are in S. If a relation is reflexive , irreflexive , symmetric , antisymmetric , asymmetric , transitive , total , trichotomous , a partial order , total order , strict weak order , total preorder weak order , or an equivalence relation , its restrictions are too. However, the transitive closure of a restriction is a subset of the restriction of the transitive closure, i.

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"; its restriction to females does relate a woman with her paternal grandmother.

Also, the various concepts of completeness not to be confused with being "total" do not carry over to restrictions. The left-restriction right-restriction , respectively of a binary relation between X and Y to a subset S of its domain codomain is the set of all pairs x , y in the relation for which x y is an element of S. Various operations on binary endorelations can be treated as giving rise to an algebraic structure , known as relation algebra. It should not be confused with relation al algebra which deals in finitary relations and in practice also finite and many-sorted.

For heterogenous binary relations, a category of relations arises. Despite their simplicity, binary relations are at the core of an abstract computation model known as an abstract rewriting system. Certain mathematical "relations", such as "equal to", "member of", and "subset of", cannot be understood to be binary relations as defined above, because their domains and codomains cannot be taken to be sets in the usual systems of axiomatic set theory.

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. Another solution to this problem is to use a set theory with proper classes, such as NBG or Morse—Kelley set theory , and allow the domain and codomain and so the graph to be proper classes: A minor modification needs to be made to the concept of the ordered triple X , Y , G , as normally a proper class cannot be a member of an ordered tuple; or of course one can identify the function with its graph in this context.

The non-symmetric ones can be grouped into quadruples relation, complement, inverse , inverse complement. From Wikipedia, the free encyclopedia. For a more general notion of relation, see finitary relation. For a more combinatorial viewpoint, see theory of relations. For other uses, see Relation disambiguation. This section may contain misleading parts. Please help clarify this article according to any suggestions provided on the talk page.

This section needs expansion with: You can help by adding to it. A relation that is reflexive, symmetric, and transitive is called an equivalence relation. A relation that is symmetric, transitive, and serial is also reflexive. A relation that is only symmetric and transitive without necessarily being reflexive is called a partial equivalence relation.

A relation that is reflexive, antisymmetric, and transitive is called a partial order. A partial order that is total is called a total order , simple order , linear order, or a chain.

In this case, if R and S disagree, R is also said to be smaller than S. If R is a binary relation over X , then each of the following is a binary relation over X:. The restriction of a binary relation on a set X to a subset S is the set of all pairs x , y in the relation for which x and y are in S. If a relation is reflexive , irreflexive , symmetric , antisymmetric , asymmetric , transitive , total , trichotomous , a partial order , total order , strict weak order , total preorder weak order , or an equivalence relation , its restrictions are too.

However, the transitive closure of a restriction is a subset of the restriction of the transitive closure, i. 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"; its restriction to females does relate a woman with her paternal grandmother. Also, the various concepts of completeness not to be confused with being "total" do not carry over to restrictions.

The left-restriction right-restriction , respectively of a binary relation between X and Y to a subset S of its domain codomain is the set of all pairs x , y in the relation for which x y is an element of S. Various operations on binary endorelations can be treated as giving rise to an algebraic structure , known as relation algebra.

It should not be confused with relation al algebra which deals in finitary relations and in practice also finite and many-sorted. For heterogenous binary relations, a category of relations arises. Despite their simplicity, binary relations are at the core of an abstract computation model known as an abstract rewriting system. Certain mathematical "relations", such as "equal to", "member of", and "subset of", cannot be understood to be binary relations as defined above, because their domains and codomains cannot be taken to be sets in the usual systems of axiomatic set theory.

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. Another solution to this problem is to use a set theory with proper classes, such as NBG or Morse—Kelley set theory , and allow the domain and codomain and so the graph to be proper classes: A minor modification needs to be made to the concept of the ordered triple X , Y , G , as normally a proper class cannot be a member of an ordered tuple; or of course one can identify the function with its graph in this context.

The non-symmetric ones can be grouped into quadruples relation, complement, inverse , inverse complement. From Wikipedia, the free encyclopedia. For a more general notion of relation, see finitary relation. For a more combinatorial viewpoint, see theory of relations. For other uses, see Relation disambiguation. This section may contain misleading parts. Please help clarify this article according to any suggestions provided on the talk page. This section needs expansion with: You can help by adding to it.

This section needs expansion. Confluence term rewriting Hasse diagram Incidence structure Logic of relatives Order theory Triadic relation. Set Theory and the Continuum Problem. Retrieved 18 November Encyclopedia of Optimization 2nd ed. A Categorical Approach to L-fuzzy Relations. The same four definitions appear in the following: Pahl; Rudolf Damrath Mathematical Foundations of Computational Engineering: Semantics of Sequential and Parallel Programs.

Modelling of Concurrent Systems: Relational Methods in Computer Science. Transactions on Rough Sets II. Lecture Notes in Computer Science. Semigroups Underlying First-order Logic. Coalgebraic Methods in Computer Science. Finite Automata, Their Algebras and Grammars: Towards a Theory of Formal Expressions.