Understanding Sets and Functions: Definitions, Properties, and Common Types

1 Sets

1.1 Definition of a Set

A set is a collection of concrete or abstract objects that share a specific property; the objects are called elements of the set.

The number of elements in a set is called its cardinality, denoted card(A). When the cardinality is finite, the set is a finite set; otherwise it is an infinite set.

1.2 Properties of Sets

Determinacy: For any element, it either belongs to the set or does not, with no ambiguity.

Uniqueness: No two elements in a set are considered the same; each element appears only once.

Unordered: All elements have equal status; the set has no inherent order.

1.3 Common Sets

Natural numbers

Integers

Positive integers

Negative integers

Rational numbers

Real numbers (including irrational numbers, which cannot be expressed as a ratio of two integers)

2 Functions

2.1 Definition of a Function

Given a set A, for each element x in A a rule assigns an element in another set B. The correspondence between elements of A and B can be expressed as a function.

3 Common Functions

3.1 Constant Functions

These functions have the same value for every input.

3.2 Linear Functions

Functions of the form f(x) = kx + b, representing a straight line.

3.3 Power Functions

Functions of the form f(x) = x^n where n is a constant exponent.

3.4 Polynomial Functions

Polynomial functions consist of a sum of multiple power functions.

3.5 Rational Functions

Defined as the ratio of two polynomial functions.

3.6 Exponential Functions

Functions where a constant base is raised to a variable exponent, e.g., f(x) = a^x.

3.7 Logarithmic Functions

Logarithmic functions are the inverses of exponential functions and follow logarithm operation rules.

3.8 Trigonometric Functions

Trigonometric functions are closely related to the unit circle. Each point on the circle can be described by an angle θ (in radians), and the coordinates are given by cosine and sine: (cos θ, sin θ).

Thus, the cosine gives the x‑coordinate and the sine gives the y‑coordinate of a point on the unit circle.

Key properties include: when θ = 0, cos θ = 1 and sin θ = 0; when θ = π/2, cos θ = 0 and sin θ = 1, etc., producing oscillating waveforms.

3 Function Graphs

Examples of graphs for the various functions are shown below:

References

https://baike.baidu.com/item/集合/2908117

https://baike.baidu.com/item/无理数

https://baike.baidu.com/item/函数/301912

https://baike.baidu.com/item/指数函数

https://baike.baidu.com/item/对数函数

https://upload.wikimedia.org/wikipedia/commons/thumb/7/72/Sinus_und_Kosinus_am_Einheitskreis_1.svg/500px-Sinus_und_Kosinus_am_Einheitskreis_1.svg.png