Binary relations and their properties
A wide range of relations on the example of setsis accompanied by a large number of concepts, beginning with their definitions and ending with an analytical analysis of paradoxes. The variety of the concept discussed in the article on the set is infinite. Although, when we talk about dual types, this means binary relations between several quantities. And also between objects or statements.
As a rule, binary relations are denoted bysymbol R, that is, if xRx for any value of x from the field R, such property is called reflexive, in which x and x are accepted objects of thought, and R serves as a sign about this or that kind of interrelation between individuals. At the same time, if we express xRy® or yRx, then this indicates a state of symmetry, where ® is a sign of implication, similar to the union "if ..., then ...." And finally, the decoding of the inscription (xRy yy Rz) ®xRz tells about the transitive relationship, and the sign of u is a conjunction.
A binary relation that happens simultaneouslyreflexive, symmetric and transitive, is called the interrelation of equivalence. The relation f is a function, and from <x, y> V f and <x, z> V f, the equality y = z follows. A simple binary function can be easily applied to two simple arguments located in a certain order, and only in this case it provides it with a value directed to these two expressions taken in a particular case.
It should be said that f maps x to y,
Proceeding from the foregoing, the relationship between the equivalence of binary relations is determined by the properties:
- reflexivity - the ratio (M ~ N);
- symmetry - if the equality M ~ N, then N ~ M;
- transitivity - if two equalities are M ~ N and N ~ P, then as a result M ~ P.
Consider the claimed properties of binary relationsmore. Reflexivity is one of the characteristics of certain relationships, where each element of the set under study is in a given equality to itself. For example, between the numbers a = c and aφ c are reflexive connections, since always a = a, c = c, aφ a, cφ c. At the same time, the ratio of the inequality a> c is antireflexive because the inequality a> a can not exist. The axiom of this property is encoded by the signs: aRc® aRa u cRc, here the symbol ® means the word "attracts" (or "implicates"), and the symbol u acts as the union "and" (or conjuncture). From this statement it follows that in the case of the truth of the judgment aRc, the expressions aRa and cRc are also true.
Symmetry leads to the presence of a relationshipand in the event that the thought objects are interchanged, that is, in the case of a symmetric relationship, the permutation of objects does not lead to a transformation of the kind of "binary relations". For example, the relation of the equality a = c is symmetric because of the equivalence of the relation c = a; the judgment of a, c is also the same, since it corresponds to the connection with a.
Transitive set is such property, atwhich satisfies the following requirement: y V x, z Î y z z Î x, where выступает is a sign replacing the words: "if ..., then ...". The formula is verbally readable in this way: "If y depends on x, z belongs to y, then z also depends on x".
