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 the concept discussed in the article on the set is infinite. Although, when talking about dual types, this means binary relationships between several values. And also between objects or statements.
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 reflexive, in which x and x are accepted objects of thought, and R serves as a sign of whether or other form of relationship between individuals. At the same time, if you 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 Ùy Rz) ®xRz tells about transitive relationship, and the sign Ù is a conjunction.
A binary relation that is both reflexive, symmetric and transitive is called an equivalence relation. The relation f is a function, and the equality y=z follows from Î f and Î f. A simple binary function can be easily appliedto two simple arguments in a certain order, and only in this case does it provide it with a meaning 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 range x and range y. However, when f extrapolates x to y, and y Í z, this causes f to show x in z. A simple example: if f(x)=2x is true for any integer x, then f is said to map the signed set of all known integers to the set of the same integers, but this time even numbers. As mentioned above, binary relations that are both reflexive, symmetric, and transitive are equivalence relationships.
Based on the above, equivalence relationships of binary relations are determined by properties:
- reflexivity - ratio (M ~ N);
- symmetries - if the equality is M ~ N, then there will be N ~ M;
- transitivity - if two equalities M ~ N and N ~ P, then as a result M ~ P.
Let's consider the declared properties of binary relations in more detail. Reflexivity is one of the characteristics of certain connections, where each element of the set under study is in a given equality to itself. For example, between the numbers a=c and a³ c there are reflexive connections, since always a=a, c=c, a³ a, c³ c. At the same time, the relation of the inequality a>c is antireflexive because of the impossibility of the existence of the inequality a>a. The axiom of this property is encoded by signs: aRc®aRa Ù cRc, here the symbol ® means the word "involves" (or "implicates"), and the sign Ù - is the union "and" (or conjunction). It follows from this statement that if the judgment aRc is true, the expressions aRa and cRc are also true.
Symmetry entails the presence of a relationship even if mental objects are interchanged, that is, with a symmetrical relationship, the rearrangement of objects does not lead to a transformation of the type "binary relations". For example, the relationship of equality a=c is symmetrical because of the equivalence of the relationship c=a; the proposition a¹c is also the same, since it corresponds to the connection with¹a.
A transitive set is a property that satisfies the following requirement: y н x, z н y ® z н x, where ® is a sign that replaces the words: "if …, then …". The formula is verbally read as follows: "If y depends on x, z belongs to y, then z also depends on x".
