1. Collection
1.1. The basic concepts and representation methods of collections
The representation method of the set:
1. Enumeration method
2. Descriptive method: Descriptive method: Use sentences (or predicate formulas) to describe element attributes.
The nature of the set:
1. Disorder, reciprocity
1.2. The relationship between sets
Inclusion relationship:
1. Definition: A and B are sets. If the elements in A are all elements in B, then B includes A, and A is included in B.
The predicate formula definition for A⊆B includes relationship: A⊆B⇔ ∀x(x∈A→x∈B)
Properties: reflexivity, transitivity, antisymmetric
relationship
1. Definition: A and B are sets. If their elements are exactly the same, then A and B are equal. Recorded as A=B.
The definition of the predicate formula of the equality relation: A=B⇔(A⊆B∧B⊆A)⇔∀x(x∈A<->x∈B)
The judgment
theorem of the set equality relation : A=B, if and only if A ⊆B and B⊆A.
There is reflexivity, transitivity and symmetry.
3.
Definition of true inclusion relationship (proper subset relationship) : A and B are sets. If ACB and A≠B, then B really includes A, or A is really included in B, and A is a proper subset of B, denoted as A⊂B.
Predicate formula definition:
A⊂B⇔(A⊆B∧A≠B)
is only transitive in nature
1.3. Special collection
Complete set E
Since any object x in the universe of discourse belongs to E, x∈E is always true.
Property: For any set A, there is A⊆E
Empty set
∅Because any object x∈∅ in the universe of discourse is a contradiction, a contradiction should be used to define ∅.
The empty set is unique
(1) because ∀x(x∈∅-→x∈A) is always true, so ∅⊆A, that is, for any set A, there is ∅⊆A.
1.4. Set operations
1.4.1. Intersection operation
The
definition of the predicate formula for the intersection operation of
A∩B set A∩B⇔(x∈A∧x∈B)
If A∩B =∅, then AB disjoint. The
properties include idempotent law, commutative law and associative law
1.4.2. Union operation
A∪B
properties: idempotent law, commutative law, associative law, identity law, zero rate, distribution law and
absorption law
1.4.3. Difference operation
The set composed of elements that belong to A but not to B is called the difference set of A and B, or the relative complement of B to A. Recorded as AB
nature:
1.4.4. Absolute complement
A set composed of elements that do not belong to A, denoted as ~A
property:
1.4.5. Symmetrical difference
A set composed of elements that belong to A but not to B or belong to B but not to A is called the symmetric difference between A and B
1.4.6. Power set
A is a set, a set composed of all subsets of A, called A's power set. Denoted as P(A)
property
1.5. Including the exclusion theorem
1.5.1. Venn diagram
Step1: Construct a Venn diagram according to known conditions;
Step2: Fill in the number of elements in the known area, and the unknown area is represented by variables;
Step3: Set up equations for unknown variables and solve;
1.5.2. Tolerance and exclusion theorem
