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写在前面
众所周之，hobee在数学的学习方面就宛如一只菜鸡，只好通过不断复习和整理的途径来巩固自己辣鸡的数学成绩。主要是我真的什么也不会了啊（哭哭脸…那就从这里开始吧！
因为考核是候题目是英文的，所以我的整理也就用英文来叭（我也不想啊…
中英术语对照表
为了防止之后我自己都看不明白，在梳理知识点之前我先写一份中英术语对照表好了。
| English | Chinese | English | Chinese | English | Chinese |
|---|---|---|---|---|---|
| Set | 集合 | element | 元素 | member of a set | 集合的成员 |
| Roster method | 枚举 | set builder notation | 集合生成公式 | empty set | 空集 |
Universal set | 全集 | Venn diagram | VN图 | set equality | 集合等价
Subset | 子集 | proper subset | 真子集 | finite set | 有限集
Infinite set | 无限集 | cardinality | 势 | power set | 幂集
Union | 并集 | Intersection | 交集 | difference | 差
Complement | 补集 | Symmetric difference | 对称差 | membership table | 成员表
Function | 函数 | domain | 值域 | codomain | 陪域
Image | 像 | preimage | 原像 | range | 值域
Onto | 满射 | One-to-one | 单射 | Surgection |
Injection | 单射 | Bijection | 双射 | Inverse | 否
Composition | 复合函数 | floor | 下取整 | ceiling | 上取整
sequence | 序列 | Geometric progression | 几何级数 | arithmetic progression | 算术级数
recurrence relation | 递推关系 | Countable set | 可数集 | Uncountable set | 不可数集
alpha null | 阿里夫零 | Binary relation from A to B | 二元关系 | relation on A | 集合A上的关系
Reflexive | 自反性 | symmetric | 对称性 | antisymmetric | 反对称性
transitive | 传递性 | n-ary relation | n元关系 | relational data model | 关系数据模型
primary key | 主键 | composite key | 复合主键 | selection operator | 选择运算符
projection | 投影 | join | 并 | digraph |
loop | 环 | closure of a relation R with respect to property | 关系R的闭包 | connective relation |
Path | 路径 | circuit | 圈 | $R^\star$ |
equivalence relation | 等价关系 | equivalent class | 等价类 | partition | 划分
Partial ordering | 偏序 | poset | 偏序集合 | comparable | 可比
total ordering | 全序 | well-ordered set | 良序集 | hasse diagram | 哈斯图
maximal element | 最大元素 | minimal element | 最小元素 | greatest element | 极大元素
least element | 极小元素 | upper bound | 上界 | lower bound | 下界
Results
Sets and Relations
Sets
A set is an unordered collection of objects.
The objects in a set are called the elements, or members of the set. $a\in A$ and $a\notin A$
A set is said to contain its elements.
Describing Sets
Roster Method
Example: $V = {a,e,i,o,u }$
Set Builder Notation
Example: $O = { x\in Z^+|x \text{ is odd and } x<10 }$
closed interval $[a,b]$
open interval $(a,b)$
Some Important Sets
$N = \text{natural numbers} = {0,1,2,3\dots}$
$Z = \text{integers} = {\dots,-3,-2,-1,0,1,2,3\dots}$
$Z^+ = \text{positive integers} = {1,2,3\dots}$
$R$ = set of real numbers
$R^+$ = set of positive real numbers
$C$ = set of complex numbers
$Q$ = set of rational numbers
Empty Set and Universal Set
Universal Set
The Universal set $U$ is the set containing everything currently under consideration.
Empty Set
The empty set is the set with no elements. Symbolized $\emptyset$ or ${}$
Notice : $\emptyset \neq {\emptyset }$
Russell’s Paradox
not so important
Subsets and Set Equality
Set Equality
Two sets are equal iff they have the same elements.
$A=B$ iff $\forall x(x\in A \leftrightarrow x\in B)$ is true
$A = B$ means $A\subseteq B$ and $B\subseteq A$
Subsets
The set $A$ is a subset of $B$ , iff every element of $A$ is also an element of $B$ .
$A\subseteq B$ iff $\forall x(x\in A \rightarrow x\in B)$ is true
Proper Subsets
If $A\subseteq B$ , but $A\neq B$ , then we say $A$ is a proper subset of $B$ .
$A\subset B$ iff $\forall x(x\in A \rightarrow x\in B) \ \wedge \exist x(x\in B\ \wedge x\notin A)$ is true
Cardinality of Sets
If there are exactly n distinct elements in $S$ where $n$ is a nonnegative integer , We say $S$ is finite, otherwise it is infinite.
The cardinality of a finite set $A$ ,denoted by $|A|$ , is the number of elements of $A$.
Power Sets
The set of all subsets of a set $A$ , denoted $P(A)$ , is called the power set of $A$.
if a set has $n$ elements , then the cardinality of the power set is $2^n$ .
Tuples
Two n-tuples are equal iff their corresponding elements are equal.
2-tuples are called ordered pairs.
Cartesian Product
The Cartesian Product of two sets $A$ and $B$ ,denoted by $A\times B$ is the set of ordered pairs $(a,b)$ where $a\in A$ and $b \in B$.
$$
A\times B = {(a,b)|a\in A\wedge b\in B}
$$
A subset $R$ of the Cartesian product $A\times B$ is called a relation from the set $A$ to the set $B$.
Truth sets of Quantifiers
Predicate $P$ and domain $D$ , the truth set of $P$ to be the set of elements in $D$ for which $P(X)$ is true. ${ x\in D|P(x)}$
Set Operations
Union
$$
A \cup B = {x|x\in A \vee x \in B}
$$
Intersection
$$
A \cap B = {x|x\in A \wedge x \in B}
$$
If the intersection is empty , then $A$ and $B$ are said to be disjoint
Complementation
$$
\overline{A} = { x\in U|x\notin A }
$$
Difference
$$
A-B = {x|x\in A\wedge x\notin B } = A\cap \overline{B}
$$
Symmetric Difference
$$
A \oplus B = (A-B)\cup (B-A)
$$
Generalized unions and intersections
$$
\bigcup^n_{i = 1} A_i = A_1 \cup A_2\cup \dots\cup A_n \
\bigcap^n_{i = 1} A_i = A_1 \cap A_2\cap \dots\cap A_n \
$$
These are well defined , since union and intersection are associative.
Set Identities
Identity laws
$$
A\cup\emptyset = A\
A\cap U = A
$$
Domination laws
$$
A\cap\emptyset = \emptyset\
A\cup U = U
$$
Idempotent laws
$$
A\cup A = A\
A\cap A = A
$$
Complementation law
$$
\overline{(\overline{A})} = A
$$
Commutative laws
$$
A\cap B = B\cap A\
A\cup B = B\cup A
$$
Associative laws
$$
A\cap (B\cap C) = (A\cap B)\cap C\
A\cup (B\cup C) = (A\cup B)\cup C
$$
Distributive laws
$$
A\cap (B\cup C) = (A\cap B)\cup (A\cap C)\
A\cup (B\cap C) = (A\cup B)\cap (A\cup C)
$$
De Morgan’s laws
$$
\overline{A\cup B} = \overline{A} \cap \overline{B}\
\overline{A\cap B} = \overline{A} \cup \overline{B}
$$
Absorption laws
$$
A \cup (A\cap B) = A\
A \cap (A\cup B) = A
$$
Complement laws
$$
A\cup \overline{A} = U\
A\cap \overline{A} = \emptyset
$$
Functions
Definition of a Function
Injection,Surjection and Bijection
Injection
A function is said to be an injection if it is one-to-one or injective.
$$
\forall a,b[f(a) = f(b)\rightarrow a =b]
$$
Surjection
A function is called a surjection if it is onto or surjective.
$$
\forall b \exist a [f(a) = b]
$$
Bijection
A function is a one-to-one correspondence, or a bijection, if it is both one-to-one and onto.
$$
\forall a,b[f(a) = f(b)\rightarrow a =b] \wedge \forall b \exist a [f(a) = b]
$$
Inverse Function
Function Composition
Partial Functions
Relations
Relations and Properties
A binary relation R from a set A to a set B is a subset $R\subseteq A\times B$
$a R b$ denotes that $(a,b) \in R$ , $a$ is related to $b$ by $R$.
A binary relation R from a set A is a subset of $A\times A$ .
Properties
Reflexive
$R$ is reflexive iff $(a,a)\in R$ for every element $a\in A$
$$
\forall x[x\in A\rightarrow(x,x)\in R]
$$
The empty relation on an empty set is reflexive vacuously.
Irreflexive
$R$ is irreflexive iff $(a,a)\notin R$ for every element $a\in A$
$$
\forall x[x\in A\rightarrow(x,x)\notin R]
$$
Symmetric
Antisymmetric
Transitive
Combining Relations
We can combine two relations using basic set operations to form new relations.
Composition
Suppose
$R_1$ is a relation from a set $A$ to set $B$ .
$R_2$ is a relation from a set $B$ to set $C$ .
Then the composition of $R_2$ with $R_1$ , is a relation from $A$ to $C$ , denoted by $R_2 \circ R_1$.
Powers of a relation
Let $R$ be a binary relation on $A$ . Then the powers $R^n$ of the relation $R$ can be defined inductively by:
$R^1 = R$ , $R^{n+1} = R^n \circ R$.
