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Binary relations and their properties

A wide range of relations on the example of sets is 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 one speaks of dual types, this means binary relations between several quantities. And also between objects or statements.

As a rule, binary relations are denoted by the symbol R, that is, if xRx for any value of x from the field R, this property is called reflexive, in which x and x are accepted objects of thought, and R serves as a sign about some form of interrelation between individuals . At the same time, if we express xRy® or yRx, then this indicates a state of symmetry, where ® is an implication sign, similar to the union "if ..., then ...." And finally, the decoding of the inscription (xRy yy Rz) ®xRz will talk about a transitive relationship, and the symbol u is a conjunction.

A binary relation, which is simultaneously reflexive, symmetric and transitive, is called the interrelation of equivalence. The relation f is a function, and from V f and 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.

We should say that f maps x to y, If f serves as a function with a zone of definition x and a zone of values of y. However, when f extrapolates x to y, and y Í z, this results in f showing x in z. A simple example: if f (x) = 2x holds true for any integer x, then we say that f maps the signed set of all known integers to the set of same integers, but this time of even numbers. As mentioned above, binary relations, which are simultaneously reflexive, symmetric and transitive, are interrelations of equivalence.

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 M ~ N and N ~ P, then as a result M ~ P.

Consider the claimed properties of binary relations in more detail. 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 links, 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 u sign 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 entails the presence of a relationship even if 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.

A transitive set is a property in which the following requirement is satisfied: y V x, z V y z z V 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".

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