#Math213

Sets

Basic

  • Concept: A set is an unordered collection of objects. The objects are called elements or members.(强调无序性)
  • Representation:
    5f83ecf312bbe59ceb88a9516063239.png
  • Subset:
    343b4685595384c6820e2df5d6a525e.png
  • Size-Carinality 集合的势
    2e5215c192361d9ad54ff5f9d50b6f1.png
  • Power Set: 集合所有子集的集合

Tuple and Cartesian Product

  • Tuple:Tuple 强调元素的有序性
    f6977c25a800ddeabe4fc532ea30dce.png

  • Cartesian Product: 各集合元素的有序组合
    a01fc501d518028c7099bc510866d82.png
  • Relation: **A subset of of the Cartesian product $A \times B$ is called a relation from the set A to the set B

Operation

  • Union:
    9316fed1ca6f06c4d93d079e6368deb.png
  • Intersection
    777a1e4fedc984c37cbddcd3905dfd7.png
  • Difference
    677aa5312aac0e76cfa348e0fd197a3.png
  • Complement
    5e7bc02206765601205e3f5289f16da.png

Identities

77fbecd60c5911d0c34e5c7f711ceab.png
9ed865c5859bce3020dcac748d5920e.png
fcc24455dca4f899efe4cdb25ceb66b.png

Function

Basic

  • Concept
    a72dd7eb3c5963abc01c8bc8a91011c.png
    domain: 定义域
    codomain: 上域
    range：值域
    $f(a)=b$: a 为 preimage, b 为 image
  • Type
    1. injective: 单射函数，不同的像对应不同的原像
      one-to-one
      ec253a79a29017fa8145b47957b9b95.png
    2. Surjective: 满射函数，对任意的像都有原像与之对应
      即陪域中每个元素都被覆盖
      onto
      a40c9e25f3ced05f54667f940291ff3.png
    3. Bijective: 双射函数，既为单射又为满射
      one-to-one correspondence
      5618eff0519eaaa9e656e1d26106ac2.png
      证明 Bijective Function: 证明单射和满射即可

c44a9da43b9349ded1c5ce5c9523068.png

  1. Inverse Function
    45e6e3184f15c0a73ea9134cf41975a.png

Composition of Functions

Concept
738f3aa9300d49fd8d4e3bee25693be.png

对于双射，映射及其逆映射的复合即为 Identity Function
ff195719aaf95196e211096d1491bb1.png

Special Function

  • Floor and Ceiling Function
  • Factorial Function

Sequence

Definition
b05311c96a40f2e7ece143e8a16bf39.png
还可利用递归形式定义
efae723fb4fdaa39b71075ffa72dea2.png
确定初始项，然后即可根据后项与前项之间的关系定义

Common Types

  • Geometric Progression: 等比
  • Arithmetic Progression: 等差

Cardinality

Countable

The elements of the set can be enumerated and listed
376caa6de6f2aa9006e9a1e678a2120.png
考察集合的势与正整数集势的关系
255eac2b42d769445cc1162e039075c.png

Core: check one-to-one correspondence 或者寻找一种合理的排列方式

Notice: 正有理数集可数，考虑控制分子分母的和，然后逐项列举
1d7a6864b584b7bbc24926ed1308649.png
17191638b8c4149e7f04443601c764d.png

注意有理数集，代数集可数

Uncountable Set

$\mathbb{R}$ , $(0,1)$ 不可数，康托尔对角线
23bcf85deb072e9b08cb5c97164b908.png

证明集合的势相等：考虑构造两者相互的 one-to-one correspondence
c33820f9a8b3d37b9a49ee7f258ee8a.png

Computable

13bf3939680722c446c2efd47571aa0.png

P: 表示 Power set,即集合所有子集构成的集合