The first element and the second element are like the x and y values of coordinates, like a dead rule.
Cartesian product is a concept that has been heard many times and often forgotten. Just like parenthesis multiplication, it is an ordered combination of items. In addition, it has four properties: first, the cartesian product of empty sets, sets and empty sets is still empty; Cartesian product does not satisfy the associative law; Cartesian product satisfies the distribution rate.
Binary relation refers to a set, generally called R, which requires that the set is empty or its elements are all ordered pairs.
In addition, a subset of A*B is called a binary relation from A to B, and if AB is equal, it is called a binary relation on A. ..
An empty relation refers to an empty set.
Global relationship refers to all the relationship components of set a.
The identity relation refers to the relation that X and Y are equal; Similarly, we can understand the less than or equal relation and the separable relation.
Relationship matrix and relationship diagram are concrete description forms of relationships, as shown in case analysis.
The second section is relational operation: domR, ranR and fldR are reinterpreted; At the same time, three relations are defined, namely, inverse, synthesis, restriction and image. And some theorems are given. Finally, the n power of the concept r is explained.
Inverse is a bit like inverse operation, pushing x from y.
Synthesis can be analogized as a composite function.
As the name implies, a constraint is a relationship under a given constraint.
Image refers to a series of relationships under given limited conditions.
The n-th power operation of R is very similar in style and strength, but it is actually a continuous composition relationship. The zeroth power of R is identity matrix.
The third section is the nature of relationship: mainly refers to five kinds, reflexivity, non-reflexivity, symmetry, antisymmetry and transitivity. First of all, it must be said that for a relationship, it may not contain any of the above properties. The following is an introduction to the characteristics based on relational matrix.
Reflexivity means that all main diagonal elements are 1.
Non-reflexivity means that the main diagonal elements are all 0.
Symmetry means that the matrix is symmetric.
Asymmetry means that two elements in a matrix that are in a symmetrical position must be 1 and the other is 0.
Transitivity means that if vertices A to B are related and B to C are related, then A to C are also related.
The fourth section is the closure of the relationship: the so-called closure and so on are all ridiculous concepts. In fact, to put it bluntly, it is to add a little element to the relationship, so that the relationship that does not have certain attributes has the desired attributes. There are three concepts, reflexive closure r(R), symmetric closure s(R) and transitive closure t(R), and some construction methods are also given.
The fifth section is about equivalence relation and partial order relation: As the name implies, it mainly introduces equivalence relation and partial order relation, in which equivalence class, quotient set, partition, partial order set, totally ordered set, Haas diagram, elements and bounds are defined.
Equivalence relation refers to the relation that satisfies reflexivity, symmetry and transitivity on nonempty set A, which can be recorded as X ~ Y. ..
Equivalence class refers to the class with complete transitivity in equivalence relation, which refers to Y and is recorded as [x].
The quotient set of A under R refers to the complete set of sibling equivalence classes under the equivalence relation R, which is recorded as A/R. ..
Division is like cutting a pie, which means dividing a set into mutually exclusive parts. Interestingly, partition and quotient set can correspond to each other.
Partial order relation refers to the relation that satisfies reflexivity, antisymmetry and transitivity on non-empty set A, which is called partial order for short and is recorded as ≤.
The poset relation R on sets A and A is collectively called poset, and is denoted as
Totally ordered set is a special case of poset. A totally ordered set is comparable to any x, y∈A, x, y, and this relationship is called total order relationship.
Hastings diagram refers to the description of posets, and its description relationship is that the lower part points to the upper part. As can be seen from the definition, totally ordered set's image is a straight line, so totally ordered set can also be called a linear ordered set.
Maximal (minimal) elements refer to the elements that all elements point to (point to all elements), and not all posets have maximal (minimal) elements.
The largest (smallest) element refers to an element that does not point to other elements (is not pointed by other elements).
The upper and lower bounds introduce a new set B. For the set B belonging to A, if there is an element Y that makes all elements in B point to Y, then Y is an upper bound of B, and the lower bound is the opposite.
As the name implies, the minimum upper bound (supremum) and the maximum lower bound (supremum) can be selected from the known upper and lower bounds.
The sixth section is the definition and properties of functions: defining functions, function values, injectivity, bijection, image of functions, constant functions, identity functions, monotone functions, characteristic functions and natural mappings.
Function and function value, I really don't want to talk about it.
Function set: If A and B are sets, all the functions from A to B constitute the set B↑A, which is pronounced as "A on B".
The image of the set under the function corresponds to the range of the function under the set (domain).
The range of surjective refers to the set B, injective refers to the one-to-one correspondence between X and Y, and injective refers to both injective and surjective.
Constant function refers to constant function, identity function refers to y=x, the abbreviation of monotone function, and characteristic function refers to the function of 0 or 1.
Natural mapping can be put forward separately, which is unfamiliar to the previous concepts. Refers to the case where an element is directly mapped to an equivalent class, such as1->; { 1,3,5}。
The seventh section is about compound function and inverse function: as shown in the title, it is completely equal to the knowledge of junior high school and senior high school, needless to say.