Reflexive: Take any element X in A, if there are both.

Reflexive: Take any element X in A, if there are both.

Representation on relation matrix:

Reflexive: All elements on the main diagonal are 1.

Reflexive: All elements on the main diagonal are 0.

Representation on the chart:

Reflexive: Every vertex has a cycle.

Reflexive: Every vertex has no cycle.

Extended data:

A summary of reflexivity and anti-reflexivity in discrete mathematics;

The order relationship in life is also the realistic embodiment of order couple. A binary group consisting of two elements x and y in a certain order is called an ordered even pair.

Because the ordered pair is ordered, the equality of the ordered pair is the equality of the corresponding elements in the corresponding position. The idea of generalized ordered pairs is to define an ordered sequence of any n elements, which can also be called an n-fold ordered group. The basic ordered pair is defined, so the relationship between sets can be defined at the level of sets.

The concept of Cartesian product can be obtained by associating ordered pairs with sets. The elements of set A X B are ordered pairs, the first element in the ordered pair is taken from A, and the second element is taken from B. At the same time, the cartesian products of sets A and B are still a set. Cartesian product does not satisfy the commutative law. The proof problem involving Cartesian product has the proof of set.

Similar to the proof of the set involved in the first chapter. The process of decomposing a topic is generally to determine the set to which the first element of an ordered pair belongs and the set to which the second element belongs, and then determine the membership relationship in the logical language according to the definitions of union operation and intersection operation, and then convert it into an equivalent expression, and then convert it back to the mathematical language according to the definition, and finally complete the proof.

The key is to use the definition of Cartesian product to determine the subordinate set. In the second chapter, there are several counting theorems to determine the number and inevitable existence of sets.

Here we begin to study the definition of relationship. Any subset of A×B is a binary relation from set a to set b, which is called relation for short.

When R is an empty set, it is an empty relation, when R is reflexive, it is an identity relation, and when this subset is equal to the original set, it is a full relation. A is the front domain of relation R, and B is the back domain of relation R. At the same time, for C and D, C is the definition domain, D is the value domain, and the union of two sets is the definition domain.

Will examine the definition domain, value domain and definition domain of the relationship, and determine the first and second elements of each relationship to solve the problem.

From the above discussion, we can know that the representation of relationships can be enumerated. At the same time, this relationship can also be expressed by directed graph and Boolean matrix. Because this relationship can be expressed by Boolean matrix, Boolean matrix can also use parallel operation and Boolean product accordingly.

At the same time, these operations also involve exchange, association and distribution. A relation is a special set with ordered pairs as elements, so all operations of the set can be used on it. Intersection, union and difference are the same as in the first chapter. Note that the complement operation is related to the Cartesian product of the original set. On this basis, the relationship can be compounded.

The essence of compound operation is synthesis, and the front and back domains are determined by intermediate elements. The processing based on relational graph should pay attention to building an intermediate bridge. Based on the relational matrix, you can directly perform Boolean product operation, and then write the answer according to the result matrix.

The related proof problem here is similar to the previous proof, and the comprehensive analysis can be completed only by introducing an intermediate variable needed for compound operation. At the same time, you should be able to cite counterexamples to illustrate the topic.

There will also be an inverse operation in the corresponding relationship. The reverse operation is to exchange the front domain and the back domain. For the transformation of relational graph, it is to change the orientation of directed graph, and for relational matrix, it is to transpose. The relevant proof continues the previous thinking. The power operation of relation is based on the principle of compound operation, and the cardinality of power set will be monotonous.

Relationships also have some special properties. Reflexivity and anti-reflexivity, symmetry and anti-symmetry, and transitivity. Reflexive, reflexive, symmetrical and antisymmetric are either black or white, and there are also cases of either/or.

Judging the self-loop of reflexive graph and the diagonal, symmetry and antisymmetry of relation matrix depends on the connection between nodes, both symmetric and antisymmetric.

Graph and matrix judgment of transitive relations may be relatively unattractive, mainly to judge whether there is an intermediate connection. The above methods are not convenient for abstract relations, and the judgment theorem on set relations is needed. Several formulas and related judgment theorems should be remembered and understood.

Relationships and close operations. The key point of closure operation is to determine whether to add the least elements and whether they have corresponding properties. Closures also have related theorem formulas. Here, we should pay attention to the proof of definitions and the use of mathematical induction.