集合与函数的基本概念及性质-优快云博客

set

set 集合

A set is an unordered collection of distinct objects, called elements or members of the set. A
set is said to contain its elements. We write a ∈ A to denote that a is an element of the set A.
The notation a ∉ A denotes that a is not an element of the set A

equal 集合的相等

Two sets are equal if and only if they have the same elements. Therefore, if A and B are sets,
then A and B are equal if and only if ∀x(x ∈ A ↔ x ∈ B). We write A = B if A and B are equal
sets

任何一个元素，如果在A中，那么一定在B中。

empty set 空集合

There is a special set that has no elements. This set is called the empty set,
or null set, and is denoted by ∅. The empty set can also be denoted by { }

空集合与 singleton set 不同。∅，和{∅}也不同，前者是空集，里面没有元素；后者是包含空集的集合，里面有一个元素。

cardinality 基数

Let S be a set. If there are exactly n distinct elements in S where n is a nonnegative integer,
we say that S is a finite set and that n is the cardinality of S. The cardinality of S is denoted
by |S|.

包含 n 个不同的元素，则基数为n。

trust set

Given a predicate P, and a domain D, we define the truth set of P to be the set of elements x in D for which P(x) is true. The truth set of P(x) is denoted by {x ∈ D ∣ P(x)}

Function 函数

函数

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.

建立从集合A到集合B的映射，只能映射到B中的一个元素。

domain, codomain, image, preimage, range,maps

If f is a function from A to B, we say that A is the domain of f and B is the codomain of f.
If f(a) = b, we say that b is the image of a and a is a preimage of b. The range, or image, of f is the set of all images of elements of A. Also, if f is a function from A to B, we say that f
maps A to B.

injection, one-to-one

A function f is said to be one-to-one, or an injection, if and only if f(a) = f(b) implies that a = b for all a and b in the domain of f. A function is said to be injective if it is one-to-one.

一一对应，每个A中元素映射的值唯一，不一定能把B中所有元素都映射到。

surjective, onto

A function f from A to B is called onto, or a surjection, if and only if for every element b ∈ B there is an element a ∈ A with f(a) = b. A function f is called surjective if it is onto.

每个B中的元素，都有A中的元素可以映射过去，但不一定用到A中所有元素。

bijection, one-to-one correspondence

The function f is a one-to-one correspondence, or a bijection, if it is both one-to-one and onto. We also say that such a function is bijective.

一一对应，且每个定义域A中的元素和值域B中的元素都被用到了。

inverse function

Let f be a one-to-one correspondence from the set A to the set B. The inverse function of f is the function that assigns to an element b belonging to B the unique element a in A such that f(a) = b. The inverse function of f is denoted by $f ^{−1}$ . Hence, $f ^{−1}(b) = a$ when f(a) = b.

前提是 f 是一一对应的。

composition

Let g be a function from the set A to the set B and let f be a function from the set B to the set C. The composition of the functions f and g, denoted for all a ∈ A by f ◦g, is the function from A to C defined by
$(f ◦ g) (a) = f (g (a))$

函数复合。

graph of function

Let f be a function from the set A to the set B. The graph of the function f is the set of ordered pairs ${(a, b) ∣ a ∈ A and f(a) = b}$ .

partial function

A partial function f from a set A to a set B is an assignment to each element a in a subset of A, called the domain of definition of f , of a unique element b in B. The sets A and B are called the domain and codomain of f , respectively. We say that f is undefined for elements in A that are not in the domain of definition of f . When the domain of definition of f equals A, we say that f is a total function

没有用到A中的所有元素，而是A的子集，被称为the domain of definition。

Cardinality of Sets

same cardinality

The sets A and B have the same cardinality if and only if there is a one-to-one correspondence from A to B. When A and B have the same cardinality, we write |A| = |B|

如果集合A，B之间可以建立一个一一映射的函数，那么就称A，B是等势的。

不等关系

If there is a one-to-one function from A to B, the cardinality of A is less than or the same as the cardinality of B and we write |A| ≤ |B|. Moreover, when |A| ≤ |B| and A and B have different cardinality, we say that the cardinality of A is less than the cardinality of B and we write |A| < |B|.

存在一个一到一的函数，从A映射到B(不一定覆盖所有B的元素)，那么就有小于等于关系。只要他俩的基数不等，那就是小于关系。

Countable Sets

A set that is either finite or has the same cardinality as the set of positive integers is called
countable. A set that is not countable is called uncountable. When an infinite set S is countable, we denote the cardinality of S by ℵ0 (where ℵ is aleph, the first letter of the Hebrew alphabet). We write |S| = ℵ0 and say that S has cardinality “aleph null.

有有限个元素的集合或者与正整数有相同基数的集合是可数的。当一个集合是可数的时候，我们用 ℵ0表示他的基数。

一个无限的集合是可数的，当且仅当它可以将元素拍成一个序列(下标是正整数)，因为一一映射的函数 f 会将正整数映射到另一个集合的元素。

所有整数的集合是可数的，可以将他按照 ${0,1,-1,2,-2,...}$ 的顺序排列。

有理数是可数的。可以将有理数看成 $\frac{p}{q}$ ，然后用列表示所有有理数中分母为q的，行表示所有有理数中分子为p的。然后按照先p+q=2的，再p+q=3的顺序排列这些有理数。具体如图，箭头表示排列的方向。如果发现 $\frac{p}{q}$ 的值出现了重复，那么就舍弃。圆圈表示被选来用来排列的那些数。
这样我们就找到了一个方法，可以按照顺序排列所有有理数，所以有理数是可数的。
在这里插入图片描述

Uncountable Set

实数是不可数集合，证明方法为 Cantor diagonalization argument，即对角线方法。
如果实数是可数的，那么0-1之间的实数应该也是可数的。那么我们假设0-1之间的实数可以按照某种顺序排列，比如：
$\begin{aligned} &r_{1}=0 . d_{11} d_{12} d_{13} d_{14} \ldots \\ &r_{2}=0 . d_{21} d_{22} d_{23} d_{24} \cdots \\ &r_{3}=0 . d_{31} d_{32} d_{33} d_{34} \cdots \\ &r_{4}=0 . d_{41} d_{42} d_{43} d_{44} \quad \ldots \\ &\quad \vdots \end{aligned}$
其中 $d_{i j} \in\{0,1,2,3,4,5,6,7,8,9\} .$ (例如，假设 $r_{1}=0.23794102 \ldots$ , 有 $d_{11}=$ $2, d_{12}=3, d_{13}=7$ 以此类推)
然后，根据下面的规则生成这样的一个实数 $d_{1} d_{2} d_{3} d_{4} \ldots$ 其中：
$d_{i}=\left\{\begin{array}{l} 4 \text { if } d_{i i} \neq 4 \\ 5 \text { if } d_{i i}=4 \end{array}\right.$
假设 $r_{1}=0.23794102 \ldots \\r_{2}=0.44590138 \ldots\\ r_{3}=0.09118764 \ldots \\ r_{4}=0.80553900 \ldots$
那么有 $d_{1} d_{2} d_{3} d_{4} \ldots=0.4544 \ldots$ , $d_{1}=4$ 因为 $d_{11} \neq 4, d_{2}=5$ 因为 $d_{22}=4, d_{3}=4$ 因为 $d_{33} \neq 4, d_{4}=4$ 因为 $d_{44} \neq 4$ 以此类推。

每个无理数都有一个独特的十进制展开，r应该也一样。但构造的这个r不等于任何的 $r_1,r_2,...$ 。假设他等于 $r_i$ ，那么理论上小数点后的每一位数都应该相同。但是如果 $r_{ii}$ 等于5，则 $r$ 中 $d_i=4$ ,所以二者不相等，矛盾。
所以r不会在上面的列表中出现，所以假设所有0-1的实数都可以被排列出来不成立。所以有理数不可数。

Result about cardinality

可数集合的并仍可数

If A and B are countable sets, then A ∪ B is also countable.

CHRODER-BERNSTEIN THEOREM

If A and B are sets with |A| ≤ |B| and |B| ≤ |A|, then |A| = |B|. In other words, if there are one-to-one functions f from A to B and g from B to A, then there is a one-to-one correspondence between A and B

Uncomputable Functions

We say that a function is computable if there is a computer program in some programming
language that finds the values of this function. If a function is not computable we say it is
uncomputable

relation

Let A and B be sets. A binary relation from A to B is a subset of A × B

关系是A和B的笛卡尔积的子集。(a,b)表示a和b之间具有关系R，R表示什么可以自己去定义。比如A表示学生，B表示课程，R表示学生选了某个课程，那么(小明，物理)就可以表示小明选了物理课。

函数可以用关系表示。如果R是关于A，B的关系，并且A的元素是有序数对的第一个元素，且唯一是，那么R也可以表示成函数的graph。二者的不同是，关系可以一对多。

relation on set

Let A and B be sets. A binary relation from A to B is a subset of A × B

Let A be the set {1, 2, 3, 4}. Which ordered pairs are in the relation R = {(a, b) ∣ a divides b}?
R = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (2, 4), (3, 3), (4, 4)}

reflexive 自反的

A relation R on a set A is called reflexive if (a, a) ∈ R for every element a ∈ A

symmetric/antisymmetric 对称/反对称的

A relation R on a set A is called symmetric if (b, a) ∈ R whenever (a, b) ∈ R, for all a, b ∈ A. A relation R on a set A such that for all a, b ∈ A, if (a, b) ∈ R and (b, a) ∈ R, then a = b is called antisymmetric
whenever：无论何时

symmetric：只要a和b满足关系，那么b和a必定满足关系。
antisymmetric：如果a和b满足关系，且b和a满足关系，那么a和b一定相等。

transitive 传递的

A relation R on a set A is called transitive if whenever (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R, for all a, b, c ∈ A

composite

Let R be a relation from a set A to a set B and S a relation from B to a set C. The composite of R and S is the relation consisting of ordered pairs (a, c), where a ∈ A, c ∈ C, and for which there exists an element b ∈ B such that (a, b) ∈ R and (b, c) ∈ S. We denote the composite of R and S by S ◦R

degree

Let A1, A2, … , An be sets. An n-ary relation on these sets is a subset of A1 × A2 × ⋯ × An. The sets A1, A2, … , An are called the domains of the relation, and n is called its degree.

selection

Let R be an n-ary relation and C a condition that elements in R may satisfy. Then the selection operator sC maps the n-ary relation R to the n-ary relation of all n-tuples from R that satisfy the condition C.

从满足关系R的元素中再选出满足条件C的。

projection

The projection $P_{i_{1} i_{2}, \ldots, i_{m}}$ where $i_{1}<i_{2}<\cdots<i_{m}$ , maps the $n$ -tuple $\left(a_{1}, a_{2}, \ldots, a_{n}\right)$ to the $m$ -tuple $\left(a_{i_{1}}, a_{i_{2}}, \ldots, a_{i_{m}}\right)$ , where $\leq n$

也就是从n元关系里面，去掉一些。原来是n个元素组成的元组，现在减少到m个了。

join

Let $R$ be a relation of degree $m$ and $S$ a relation of degree $n .$ The join $J_{p}(R, S)$ where $\leq m$ and $\leq n$ , is a relation of degree $m + n - p$ that consists of all $(m + n - p)$ -tuples $\left(a_{1}, a_{2}, \ldots, a_{m-p}, c_{1}, c_{2}, \ldots, c_{p}, b_{1}, b_{2}, \ldots, b_{n-p}\right), \quad$ where the $m$ -tuple $\left(a_{1}, a_{2}, \ldots, a_{m-p}, c_{1}, c_{2}, \ldots, c_{p}\right)$ belongs to $R$ and the $n$ -tuple $\left(c_{1}, c_{2}, \ldots, c_{p}, b_{1}, b_{2}, \ldots, b_{n-p}\right)$ belongs to $S$ .

也就是R和S中有公共部分(c相关的)，然后找出来有公共部分的所有组合。

closure

If R is a relation on a set A, then the closure of R with respect to P, if it exists, is the relation S on A with property P that contains R and is a subset of every subset of A × A containing R with property P.

也就是满足关系R，同时具有性质P。

equivalence relation

A relation on a set A is called an equivalence relation if it is reflexive, symmetric, and transitive.

equivalent

Two elements a and b that are related by an equivalence relation are called equivalent. The notation a ∼ b is often used to denote that a and b are equivalent elements with respect to a particular equivalence relation

如果a和b在某个等价关系下存在关系，那么就说a和b是equivalent的。a ~ b 表示在某个equivalence关系下a和b是equivalent的。

equivalence class

Let $R$ be an equivalence relation on a set $A .$ The set of all elements that are related to an element $a$ of $A$ is called the equivalence class of $a$ . The equivalence class of $a$ with respect to $R$ is denoted by $a]_{R} .$ When only one relation is under consideration, we can delete the subscript $R$ and write $[a]$ for this equivalence class.
$[a]_{R}=\{s \mid(a, s) \in R\}$

和a存在关系的所有元素构成的集合被称为等价类。

Theorem 等价关系

Let $R$ be an equivalence relation on a set $A .$ These statements for elements $a$ and $b$ of $A$ are equivalent:
(i) $a R b$
(ii) $[a] = [b]$
(iii) $\cap[b] \neq \emptyset$

partial ordering

A relation R on a set S is called a partial ordering or partial order if it is reflexive, antisymmetric, and transitive. A set S together with a partial ordering R is called a partially ordered set, or poset, and is denoted by (S, R). Members of S are called elements of the poset.

偏序关系：自反的，反对称的，传递的。

comparable

The elements $a$ and $b$ of a poset $\preccurlyeq)$ are called comparable if either $\preccurlyeq b$ or $\preccurlyeq a$ . When $a$ and $b$ are elements of $S$ such that neither $\preccurlyeq b$ nor $\preccurlyeq a, a$ and $b$ are called incomparable.

比如在偏序集 $Z^+,|)$ 中，| 表示整除关系。3和9是课比较的，因为3整除9，但是5和7不是可比较的，因为5不整除7，7也不整除5.

linearly ordered set， chain

If $\preccurlyeq)$ is a poset and every two elements of $S$ are comparable, $S$ is called a totally ordered or linearly ordered set, and $\preccurlyeq$ is called a total order or a linear order. A totally ordered set is also called a chain.

偏序集合中任何两个元素都是可比的，那么就可以把这个集合称为先行顺序集合\全序。

$(Z^+,\leq)$ 就是全序的，因为集合中任何两个数a，b要么a≤b，要么b≤a

well-ordered

$\preccurlyeq)$ is a well-ordered set if it is a poset such that $\preccurlyeq$ is a total ordering and every nonempty subset of $S$ has a least element.

lattice

A partially ordered set in which every pair of elements has both a least upper bound and a
greatest lower bound is called a lattice.

boolean variable, boolean function

Let $B=\{0,1\} .$ Then $B^{n}=\left\{\left(x_{1}, x_{2}, \ldots, x_{n}\right) \mid x_{i} \in B\right.$ for $\left.1 \leq i \leq n\right\}$ is the set of all possible $n$ -tuples of $\mathrm{~s}$ and $\mathrm{~s}$ . The variable $x$ is called a Boolean variable if it assumes values only from $B$ , that is, if its only possible values are 0 and $1 .$ A function from $B^{n}$ to $B$ is called a Boolean function of degree $n .$

boolean expression

Boolean functions can be represented using expressions made up from variables and Boolean operations. The Boolean expressions in the variables $x_{1}, x_{2}, \ldots, x_{n}$ are defined recursively as
$x_{1}, x_{2}, \ldots, x_{n}$ are Boolean expressions;
if $E_{1}$ and $E_{2}$ are Boolean expressions, then $\bar{E}_{1},\left(E_{1} E_{2}\right)$ , and $\left(E_{1}+E_{2}\right)$ are Boolean expressions.