A wide range of relations on the example of sets is accompanied by a large number of concepts, starting with their definitions and ending with an analytical analysis of paradoxes. The variety of concepts discussed in the article on the set is infinite. Although, when talking about dual types, this means binary relations between several quantities. And also between objects or utterances.
As a rule, binary relations are denoted by the symbol R, that is, if xRx for any value x from the field R, such a property is called reflective, in which x and x are accepted objects of thought, and R serves as a sign of some form of relationship between individuals . At the same time, if xRy® or yRx is expressed, this indicates a state of symmetry, where ® is the implication sign, similar to the union “if ... then ...”. And, finally, the decoding of the inscription (xRy Ùy Rz) ®xRz talks about the transitive relationship, and the sign Ù is a conjunction.
The binary relation, which is both reflexive, symmetric, and transitive, is called the equivalence relation. The relation f is a function, and the equality y = z follows from <x, y> Î f and <x, z> Î f. A simple binary function can be easily applied to two simple arguments arranged 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,
if f is a function with a zone of definition of x and a zone of values of y. However, when f extrapolates x to y, and y Í z, this leads to the fact that f shows x in z. A simple example: if f (x) = 2x is valid for any integer x, then they say that f maps the signed set of all known integers to the set of the same integers, but this time even numbers. As mentioned above, binary relations, which are simultaneously reflexive, symmetrical and transitive, are interrelations of equivalence.
Based on the foregoing, the relationship of the equivalence of binary relations is determined by the properties:
- reflexivity - ratio (M ~ N);
- symmetries - if the equality is M ~ N, then there will be N ~ M;
- transitivity - if two equalities are M ~ N and N ~ P, then the result is M ~ P.
Consider the declared properties of binary relations in more detail. Reflexivity is one of the characteristics of some relationships, where each element of the studied set is in this equality to itself. For example, between the numbers a = c and a ³ c there are reflective connections, since always a = a, c = c, a ³ a, c ³ s. At the same time, the ratio of the inequality a> c is antireflective due to the impossibility of the existence of the inequality a> a. The axiom of this property is encoded by the signs: aRc® aRa Ù cRc, here the symbol ® means the word "attracts" (or "implicates"), and the sign Ù - stands for the union "and" (or conjunction). 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 mental objects are interchanged, that is, with a symmetrical relationship, the rearrangement of the objects does not lead to a transformation of the form 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 a¹c is also the same, since it corresponds to a connection with a¹a.
A transitive set is such a property that satisfies the following requirement: y Î x, z Î y ® z Î x, where ® acts as a sign replacing the words: "if ... then ...". The formula is read verbally in this way: "If y depends on x, z belongs to y, then z also depends on x."