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(像)
- 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