# 数学代写|抽象代数作业代写abstract algebra代考|Equivalence Relations

#### Doug I. Jones

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## 数学代写|抽象代数作业代写abstract algebra代考|A Smidgeon of Set Theory

There is a vast formal theory of sets, but for our purposes, it suffices to informally view a set as a collection of elements.

Definition 1.7. A set is a (possibly empty) collection of elements. Thus if $S$ is a set, then each object $a$ either is an element of $S$ or is not an element of $S$. We write
$a \in S$ if $a$ is an element of $S$, and $a \notin S$ if not.

The empty set, denoted $\emptyset$, is the set containing no elements; i.e., for every object $a$ we have $a \notin S$. If $S$ is a finite set, we write $# S$, or sometimes $|S|$, to denote the number of elements that it contains.
Example 1.8. We can describe a set by explicitly listing its elements, for example,
$${1,2,3}, \quad{-11,23,19}, \quad \text { Alice, Bob, Carl }} .$$
We can describe a set by a rule, for example
$$\text { {positive integers } n \quad: \quad n \text { is a multiple of } 5} .$$
Read this colon as “such that.”
Sometimes the rule is implicitly described via a pattern, so this last example might be written as
$${5,10,15,20, \ldots}$$
This means that you, the reader, are then expected to intuit that the dots mean to continue adding multiples of 5 .

Definition 1.9. A very important example of a set is the set of natural numbers, defined informally by
$$\mathbb{N}={\text { natural numbers }}={1,2,3,4, \ldots}$$
More formally, the set of natural numbers is created as follows: ${ }^6$
(1) $\mathbb{N}$ contains an initial element 1 .
(2) For each element $n \in \mathbb{N}$, there is an increment rule that creates the next element $n+1$.
(3) It is possible to reach every element of $\mathbb{N}$ by starting with 1 and repeatedly applying the increment rule.

We say that $m$ is less than $n$ if $m$ appears before $n$ when we start at 1 and repeatedly apply the increment rule. In this case we write $m<n$. If we also want to allow for $m$ and $n$ to be equal, we write $m \leq n$. The natural numbers are an example of a totally ordered $\mathrm{set}^7,^7$ because for every pair of elements $m$ and $n$, we always have either $m \leq n$ or $n \leq m$. (Note that this “or” is inclusive, not exclusive!)

## 数学代写|抽象代数作业代写abstract algebra代考|Equivalence Relations

It is often convenient to split a set into a union of disjoint subsets and then to view the elements in each subset as being “identical” or “equivalent.” For example, consider the set of animals
$S={$ cat, lizard, dog, ant, elephant, whale, trout, mosquito, bat $}$.

We formalize this idea, which leads to a key concept that is ubiquitous in the study of algebraic systems.

Definition 1.21. Let $S$ be a set, and let $S_1, \ldots, S_n$ be subsets of $S$. We say that $S$ is the disjoint union of $S_1, \ldots, S_n$ if
$$S=S_1 \cup \cdots \cup S_n \quad \text { and } \quad S_i \cap S_j=\emptyset \text { for every } i \neq j .$$
In other words, the set $S$ is the disjoint union of the subsets $S_1, \ldots, S_n$ if every element of $S$ is in exactly one of the subsets.

As described earlier, if $S$ is the disjoint union of $S_1, \ldots, S_n$, it is often convenient to view the elements in each $S_i$ as being equivalent to one another. Turning this around, we can start with a set $S$ and describe the properties that such an “equivalence” should have.
Definition 1.22. Let $S$ be a set. An equivalence relation on $S$ is a set of ordered pairs
$$\mathcal{R} \subseteq S \times S$$
with the following three properties:
Reflexive Property: $\quad(a, a) \in \mathcal{R} \quad$ for all $a \in S$.
Symmetry Property: $\quad(a, b) \in \mathcal{R} \Longleftrightarrow(b, a) \in \mathcal{R} \quad$ for all $a, b \in S$.
Transitive Property: $\quad(a, b) \in \mathcal{R}$ and $(b, c) \in \mathcal{R} \Longrightarrow(a, b) \in \mathcal{R} \quad$ for all $a, b, c \in S$.
We then define elements $a, b \in S$ to be $\mathcal{R}$-equivalent if $(a, b) \in \mathcal{R}$, and otherwise we say that $a$ and $b$ are $\mathcal{R}$-inequivalent. We use the symbol sym $_{\mathcal{R}}$, so just sym if $\mathcal{R}$ has been specified, to indicate equivalence. Thus
$$a \sim b \text { if }(a, b) \in \mathcal{R} \quad \text { and } \quad a \not b \text { if }(a, b) \notin \mathcal{R} \text {. }$$

## 数学代写|抽象代数作业代写abstract algebra代考|A Smidgeon of Set Theory

${1,2,3}, \backslash q u a d{-11,23,19}, \backslash q u a d ~ \backslash t e x t ~{$ Alice, Bob, Carl $}}$ 。

{positive integers $} \mathrm{n} \backslash$ quad: $\backslash$ quad $\mathrm{n} \backslash$ text ${$ is a

$$5,10,15,20, \ldots$$

$$\mathbb{N}=\text { natural numbers }=1,2,3,4, \ldots$$

(1) $\mathbb{N}$ 包含初始元素 1 。
(2) 对于每个元素 $n \in \mathbb{N}$, 有创建下一个元素的递增规则 $n+1$. 则。

## 数学代写|抽象代数作业代写abstract algebra代考|Equivalence Relations

$S=\$ c a t$, lizard, dog, ant, elephant, whale, trout, mosq 我们将这个想法形式化，这导致了一个在代数系统研究 中普遍存在的关键概念。 定义 1.21。让$S$是一个集合，让$S_1, \ldots, S_n$是子集$S$. 我们说$S$是的不相交联合$S_1, \ldots, S_n$如果$S=S_1 \cup \cdots \cup S_n \quad$and$\quad S_i \cap S_j=\emptyset$for every$i \neq$换句话说，集合$S$是子集的不相交并集$S_1, \ldots, S_n$如果 每个元素$S$恰好在其中一个子集中。 如前所述，如果$S$是的不相交联合$S_1, \ldots, S_n$，通常很 方便查看每个中的元素$S_i$相当于彼此。反过来，我们可 以从一组开始$S$并描述这种“等价”应该具有的性质。 定义 1.22。让$S$是一个集合。上的等价关系$S$是一组有 序对 $$\mathcal{R} \subseteq S \times S$$具有以下三个属性: 自反属性:$(a, a) \in \mathcal{R} \quad$对全部$a \in S$. 对称性:$\quad(a, b) \in \mathcal{R} \Longleftrightarrow(b, a) \in \mathcal{R} \quad$对全部$a, b \in S$. 传递属性:$\quad(a, b) \in \mathcal{R}$和$(b, c) \in \mathcal{R} \Longrightarrow(a, b) \in \mathcal{R} \quad$对全部$a, b, c \in S$. 然后我们定义元素$a, b \in S$成为$\mathcal{R}$-等价于$(a, b) \in \mathcal{R}$，否则我们说$a$和$b$是$\mathcal{R}$-不等价。我们使用符号$\operatorname{sym}_{\mathcal{R}}$， 所以只是 sym 如果$R$已指定，以表示等价。因此$a \sim b$if$(a, b) \in \mathcal{R} \quad$and$\quad a b$if$(a, b) \notin \mathcal{R}\$.

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MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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