On the concept of set;
Concepts such as point, line and surface are primitive and undefined in geometry, while set is primitive and undefined in set theory.
Junior high school algebra, once understood "the set of positive numbers" and "the set of inequality solutions"; It is also known in junior high school geometry that the vertical line is "a collection of points with equal distance to two fixed points" and so on. When we came into contact with the concept of set, we got a preliminary understanding of the concept mainly through examples. The sentence "Generally, some specified objects are gathered together to form a set, also called a set" given in the textbook is only a descriptive explanation of the concept of set.
We can cite many practical examples in life to further illustrate this concept, so as to clarify that the concept of set, like other mathematical concepts, is not imagined by people, but comes from the real world.
In short, set: to put some specified objects together to form a set.
Representation method of set
1. enumeration: a method of listing the elements in a collection one by one and writing them in braces to represent the collection.
For example, by equation
The set of all solutions of can be expressed as {- 1, 1}.
Note: (1) Some sets can also be expressed as:
Set of all integers from 5 1 to 100: {5 1, 52,53, …, 100}
The set of all positive odd numbers: {1, 3, 5, 7, …}
(2)a is different from {a}: A represents an element, {a} represents a set, and a set has only one element.
Description: A method of indicating whether an object belongs to this set with a certain condition and writing this condition in braces to indicate the set.
Format: {x∈a|
p(x)}
Meaning: Set X that satisfies the condition p(x) in Set A.
For example, inequality
The solution set of can be expressed as:
or
The set of all right triangles can be expressed as:
Note: (1) The vertical line and the left part can be omitted without confusion.
Such as: {right triangle}; {Real number greater than 104}
(2) Error expression: {real number set}; {All Real Numbers}
3. Venn diagram: a method of representing a set with the interior of a closed curve.
Note: When to use enumeration? When to use descriptive methods?
( 1)
The common attributes of some sets are not obvious, so it is difficult to generalize them and it is not convenient to express them by description. Only enumeration can be used.
(2)
Some elements in the collection cannot be listed one by one, or it is inconvenient and unnecessary to list them one by one. Description is often used.