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# 数学代写|抽象代数代写Abstract Algebra代考|Binary operations

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## 数学代写|抽象代数代写Abstract Algebra代考|Binary operations

When one begins to think on the variety of equations one tries to solve, it doesn’t take long to realize that each subject has its own objects, upon which are defined a variety of processes that you can use to manipulate them: addition on numbers, matrix multiplication on matrices, etc. But despite the apparent differences between them, they all share one fundamental property: whenever you combine two or more of the objects, you get another object of the same type: the sum of two numbers is a number, the product of two $n \times n$ matrices is another matrix, etc. Hence, that’s where we’ll start our study of abstract algebra proper.

To begin, let’s reflect on what we’re really doing when we “combine” objects. We take two objects from a set and apply some kind of rule or process to this pair of objects, and the result produces an object from that same set. Yet what we’ve described here is nothing more than a function whose domain is ordered pairs of elements from a set and whose range is contained back in that same set. This basic observation is codified in the following definition.

Definition 2.1. Let $G$ be a set. Any function, $$, whose domain is G \times G is a binary operation on G if and only if the range of$$ is a subset of $G$. When $$is a binary operation on G, we say that G is closed under$$. We denote the image of $(a, b)$ under $*$ as $a * b$. We write $\langle G, *\rangle$ to indicate that * is a binary operation on $G$, and we say that $\langle G, *\rangle$ is a binary structure.

Although a binary operation is simply a particular function, it’s probably best to review the terms about functions and how they apply to this definition. If you need, read Section A.2 in Appendix A for a refresher on the relevant terminology.
(1) If $*$ is a binary operation on a set $G$, then the element $a * b$ cannot be undefined for any elements $a, b \in G$. For instance, if we let $G=\mathbb{Q}$ and define $\frac{a}{b} * \frac{c}{d}=\frac{a}{b} / \frac{c}{d}$, then this rule isn’t defined when $c=0$, since division by zero isn’t defined.

(2) Likewise, if $*$ is a binary operation on a set $G$, then the element $a * b$ must be welldefined for each pair of elements $a, b \in G$. For instance, if we let $G=\mathbb{Q}$ and define $\frac{a}{b} * \frac{c}{d}=a+c$, then this rule isn’t well-defined, since $\frac{1}{2}$ and $\frac{2}{4}$ represent the same number, but $\frac{a}{b} * \frac{1}{2}=a+1$, and $\frac{a}{b} * \frac{2}{4}=a+2$, which are always different.

Aside. Students often fail to appreciate the import of this example. The issue of a welldefined function always arises whenever the objects in the set have more than one description or form. The standard way to verify that a binary operation $*$ is well-defined is to take two equivalent forms of the objects in your set and prove that the result does not depend on the form of the objects. Functions dealing with rational numbers frequently fall in this category, since every rational number has infinitely many equivalent fractional forms. Hence, to check that a function $f$ is well-defined in this case, you would need to suppose that $\frac{a}{b}=\frac{c}{d}$ and prove that $f\left(\frac{a}{b}\right)=f\left(\frac{c}{d}\right)$.

## 数学代写|抽象代数代写Abstract Algebra代考|Binary tables

When a set $G$ is small, one way to define a binary operation $*$ on $G$ is to list all of the possible combinations of $a * b$ for elements $a, b \in G$. The most convenient way to do this is to set up a table whose entries indicate the result of applying the binary operation to two elements. Specifically, we create a binary table to define a binary operation on the set $G=\left{a_1, \ldots, a_n\right}$ in the following way:
1) Write the elements of $G$ in a column, then write them in a row at the top in the same order as you did in the column.
2) Fill in the corresponding $n^2$ entries with exactly one element in $G$.
3) Define the element $a_i * a_j$ to be the entry in the $i^{\text {th }}$ row and the $j^{\text {th }}$ column.
It’s also easy to verify that your table gives a rule that is both well-defined and is defined everywhere. After all, as long as every entry is filled with at least one element of $G$, then $a * b$ is defined everywhere; and as long as you don’t put more than one element from $G$ in any entry, then your rule is well-defined.

Exercise 2.9. Let $G={a, b, c}$. Suppose we define a binary operation on $G$ with the following table:
\begin{tabular}{|c||c|c|c|}
\hline$*$ & $a$ & $b$ & $c$ \
\hline \hline$a$ & $a$ & $c$ & $c$ \
\hline$b$ & $b$ & $b$ & $b$ \
\hline$c$ & $a$ & $a$ & $b$ \
\hline
\end{tabular}
(So, for instance, $c * a=a$.)
(1) Compute $a * b, \quad b * a, \quad b * b, \quad(a * c) * b, \quad a *(c * b), \quad c *(c * c)$, and $(c * c) * c$.
(2) Determine if the operation is commutative, associative, neither, or both.

There’s really no theory about binary tables, but there are several observations that are useful to have. First, since you can put any of the $n$ elements of $G$ into any of the $n^2$ entries, there are a total of $n^{n^2}$ possible ways to construct a binary operation on $G$. Second, checking to see if a binary table’s operation is commutative is easy: reflect the table along the “main diagonal” and compare with the original table. If they’re the same, then it’s a commutative operation; otherwise, you’ll have at least one pair of elements $a_i, a_j$ such that $a_i * a_j \neq a_j * a_i$.

Associativity, on the other hand, is never easy to check by looking at the table. That’s because checking associativity deals with using the table sequentially. It also means verifying that $(a * b) * c=a *(b * c)$ for all choices of $a, b, c \in G$, which means you’ve got $n^3$ different pairs of triples to compare. That’s just too tedious to do by hand; a computer is almost a necessity if you need to know if your operation is associative.

## MATLAB代写

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。