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# 数学代写|抽象代数代写Abstract Algebra代考|MATH413 Functions

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## 数学代写|抽象代数代写Abstract Algebra代考|Domain and codomain

Example 15.1. Consider the function $\gamma: \mathbb{Z}{12} \rightarrow \mathbb{Z}{18}$ where $\gamma(a)=6 a$ for all $a \in \mathbb{Z}_{12}$.

The domain of the function is $\mathbb{Z}_{12}$. This is the set of all possible inputs into the function.

The codomain of the function is $\mathbb{Z}_{18}$. This set contains all outputs (and possibly other elements).

The rule of the function is $\gamma(a)=6 a$. Here, $a$ is in the domain $\mathbb{Z}{12}$ and $\gamma(a)$ is in the codomain $\mathbb{Z}{18}$.

For instance, we have $\gamma(5)=6 \cdot 5=30=12$, where the computation $30=12$ was done in $\mathbb{Z}{18}$. Similarly, $\gamma(8)=6 \cdot 8=48=12$, where the computation $48=12$ occurs in $\mathbb{Z}{18}$. Thus, the elements 5 and 8 in the domain $\mathbb{Z}_{12}$ both map to the same element 12 in the codomain $\mathbb{Z}_{18}$. This is depicted in the diagram below:

Example 15.2. Consider the function $\varphi: \mathbb{Z} \rightarrow \mathbb{Z}_5$ where $\varphi(a)=a(\bmod 5)$ for all $a \in \mathbb{Z}$. For instance, we have $\varphi(43)=3$, since $43=3(\bmod 5)$. Similarly, $\varphi(-14)=1$, because $-14=1(\bmod 5)$. The domain of the function is $\mathbb{Z}$ and its codomain is $\mathbb{Z}_5$. The rule of the function is $\varphi(a)=a(\bmod 5)$.

## 数学代写|抽象代数代写Abstract Algebra代考|One-to-one function

Example 15.4. Consider again the function $f: U_{35} \rightarrow U_{35}$ where $f(x)=3 x$ for all $x \in U_{35}$. Choose two different inputs from the domain $U_{35}$, say $a=8$ and $b=22$. Their corresponding outputs are $f(a)=24$ and $f(b)=31$, which are in the codomain $U_{35}$. These outputs are different from each other as well.

In fact, the following is true for all $a, b \in U_{35}$ : If $a \neq b$, then $f(a) \neq f(b)$; i.e., different inputs map to different outputs. We can verify this by computing $f(x)$ for every $x \in U_{35}$ and seeing that all the outputs are different. Rather than taking this tedious approach (after all, there are 24 elements in $U_{35}$ ), we will prove this implication in a way that can be generalized to other scenarios.

Specifically, we will prove its contrapositive; namely: If $f(a)=f(b)$, then $a=b$.
Assume $f(a)=f(b)$, where $a, b \in U_{35}$. Then $3 a=3 b$ in $U_{35}$. The multiplicative inverse of 3 is 12 , where $3 \cdot 12=1$ and $12 \cdot 3=1$ modulo 35 . Multiplying both sides of the equation $3 a=3 b$ by 12 , we obtain $12 \cdot(3 a)=12 \cdot(3 b)$. Thus $(12 \cdot 3) \cdot a=(12 \cdot 3) \cdot b$, which implies $1 \cdot a=1 \cdot b$. Therefore, $a=b$.
Example $15.4$ illustrates the notion of a one-to-one function, which is defined below.

. . . .

## 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。