I. Collection of related concepts

1, meaning of set: some specified objects are set together into a set, where each object is called an element.

2. Three characteristics of elements in a set:

1. element determinism; 2. Mutual anisotropy of elements; 3. The disorder of elements

Description:

(1) For a given set, the elements in the set are certain, and any object is either an element of the given set or not.

(2) In any given set, any two elements are different objects. When the same object is contained in a set, it has only one element.

(3) The elements in the set are equal and have no order. So to judge whether two sets are the same, we only need to compare whether their elements are the same, and we don't need to examine whether the arrangement order is the same.

(4) The three characteristics of set elements make the set itself deterministic and holistic.

3. Representation of set: {? Such as {basketball players in our school}, {Pacific Ocean, Atlantic Ocean, Indian Ocean, Arctic Ocean}

Use Latin letters to represent the set: a={ basketball players in our school}, b={ 1, 2, 3, 4, 5}

Representation of collections: enumeration and description.

Note: Commonly used number sets and their symbols:

The set of nonnegative integers (i.e. natural number set) is recorded as n.

Positive integer set n* or n+ integer set z rational number set q real number set r

On the concept of "belonging"

Elements in a collection are usually represented by lowercase Latin letters. For example, if A is an element of set A, it means that A belongs to set A, and it is marked as A ∈ A; On the contrary, if a does not belong to the set a, it is recorded as a(a

Enumeration: enumerate the elements in the collection one by one, and then enclose them in braces.

Description: A method of describing the common attributes of elements in a collection and writing them in braces to represent the collection. A method to indicate whether some objects belong to this set under certain conditions.

① Language Description: Example: {A triangle that is not a right triangle}

② Description of mathematical formula: If the solution set of inequality x-3]2 is {x(r| x-3]2} or {x| x-3]2}.

4, the classification of the set:

1. The finite set contains a set of finite elements.

2. An infinite set contains an infinite set of elements.

3. An example of an empty set without any elements: {x|x2=-5}

Second, the basic relationship between sets

"Containment" Relation-Subset

Note: There are two possibilities that A is a part of B (1); (2)a and B are the same set.

On the other hand, set A is not included in set B, or set B does not include set A, which is marked as ab or ba.

"Equality" relation (5≥5 and 5≤5, then 5=5)

Example: let a = {x | x2-1= 0} b = {-1,1} "The elements are the same".

Conclusion: For two sets A and B, if any element of set A is an element of set B and any element of set B is an element of set A, we say that set A is equal to set B, that is, A = B.

(1) Any set is a subset of itself. one

② proper subset: If a(b) and a( b), then set A is the proper subset of set B, and it is denoted as ab (or ba).

③ If a(b, b(c), then a(c

(4) if a(b) and b(a), then a = b.

3. A set without any elements is called an empty set and recorded as φ.

It is stipulated that an empty set is a subset of any set and an empty set is a proper subset of any non-empty set.