Chaptcer 02 Basic Structures 基本结构

Part 02 Function and Sequence

Covering 2.3~2.4


1. Functions 函数

1.1. Definitions 定义

Let A and B be nonempty sets. A function f from A to B is an assignment of exactly one element of B to each element of A.

We write f (a) = b if b is the unique element of B assigned by the function f to the element a of A.

If f is a function from A to B, we write f : A → B.

A → B is a subset of A × B

Functions are sometimes also called mappings映射) or transformations变换

  • A : domain定义域

  • B : codomain陪域

  • b : image

  • a : preimage原像

The range值域), or image, of f is the set of all images of elements of A.

If f is a function from A to B, we say that f maps映射) A to B.

Two Notations:

Let f be a function from A to B and let S be a subset of A. The image of S under the function f is the subset of B that consists of the images of the elements of S. We denote the image of S by f (S), so

Explaination:

For the first equivalent, just prove and using the method proof by cases

For the second one, itself is easy to prove, but the interesting part is how to give a counterexample to prove that is wrong.

This counterexample is that when f() is not one-to-one (this will be mentioned right below), the latter one can be larger than the previous one! (Specific example omitted)

The graph(图) of the function f is the set of ordered pairs

A monotonic单调) function f is either monotonically (strictly) increasing () or monotonically (strictly) decreasing (())

1.2. Correspondences 对应关系

1.2.1. One-to-One Function 一对一函数

A function f is one-to-one一对一) (denoted 1-1), or injective单射函数)if

1.2.2. Onto Functions 映上函数

A function f from A to B is called onto映上), or surjective满射) if

In short, every b in B has a preimage.

1.2.3. One-to-one Correspondence Functions 一一对应函数

The function f is a one-to-one correspondence一一对应), or a bijection双射), if it is both one-to-one and onto

Whenever there is a bijection from A to B, the two sets must have the same number of elements or the same cardinality.

1.2.4. Examples of Different Types of Correspondences

Suppose that f : A → B

  • To show that f is injective, show that if f (x) = f (y) for arbitrary x, y ∈ A, x = y.
  • To show that f is not injective, find particular elements x, y ∈ A such that x ≠ y and f (x) = f (y).
  • To show that f is surjective, consider an arbitrary element y ∈ B and find an element x ∈ A such that f (x) = y.
  • To show that f is not surjective Find a particular y ∈ B such that f (x) ≠ y for all x ∈ A.

1.3. Special Functions 特殊函数

1.3.1. Inverse Functions 反函数

Let f be a one-to-one correspondence from the set A to the set B, the inverse function of f is denoted by

iff

No inverse function exists unless f is a bijection

Function f is invertible可逆的) iff f is bijective

1.3.2. Compositions of Functions 复合函数

$f\circ g$ can’t be defined unless the range of g is a subset of the domain of f.

1.4. Two Important Functions 两大重要函数

1.4.1. The floor function 取底函数

The floor function f (x) is the largest integer less than or equal to the real number x

The floor function is often also called the greatest integer function. It is often denoted by [x]

1.4.2. The ceiling function 取顶函数

2. Sequence 序列

A sequence is a function from a subset of the set of intergers (usually either the set {0,1,2,…} or the set {1,2,3,…}) to a set S. We use the notation to denote the image of the image of the integer n. We call a term of the sequence .

The order in a sequence matters!

2.1. Some Sequences 一些序列

2.1.1. Geometric Progression 几何级数

2.1.2. Arithmetic Progression 算术级数

2.1.3. Recurrence 递推

e.g. Fibonacci Sequence

2.2. Summations 求和

2.2.1. Some Useful Summation Formulae 求和公式

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