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# 数学代写|离散数学代写Discrete Mathematics代考|CSC226 Terminology

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## 数学代写|离散数学代写Discrete Mathematics代考|Terminology

Before embarking on the introduction of some of the methods of proof, it is important to introduce some terminology related to proofs:

Definition: A statement expressing the essential nature of a concept and a set of associated properties that describe the concept.
Axiom: A self-evident true statement, that is, a statement that is accepted on its intrinsic merit without proof. It may also be known as postulate.
Theorem: A mathematical statement that can be shown (proven) to be true.
Corollary: A proposition that can be proven as an immediate consequence of some other theorems.
Lemma: A less important theorem that can help prove a more important theorem.

Conjecture: A statement that is being proposed to be a true statement but is not proven yet.
Example $4.1$
Provide specific examples to highlight some terminology related to proofs.
Solution
Definition: A circle is a closed plane curve every point of which is equidistant from a fixed point within the curve.
Axiom: In Euclidean geometry, within a two-dimensional plane, for every given straight line and a point that is not on the line, there exists exactly one straight line passing through the point that is parallel to the line.
Theorem: If two sides of a triangle are equal, then the angles opposite them are equal.
Corollary: If three sides of a triangle are equal, then all three angles of the triangle are equal.
Lemma: If we subtract 1 from a positive integer, then the result is either a positive integer or 0 .
Conjecture: If a transformation sends an even integer $x$ to $\frac{x}{2}$ and an odd integer $x$ to $3 x+1$, then for all positive integers $x$, the repeated application of the transformation will eventually reach integer 1.

## 数学代写|离散数学代写Discrete Mathematics代考|Proofs of Equivalence

Sometimes a theorem states that a group of $n \geq 2$ propositions $p_1, p_2, \ldots, p_n$ are equivalent $\left(p_1 \leftrightarrow p_2 \leftrightarrow \ldots \leftrightarrow p_n\right)$, that is, they have the same truth values. To show a proof of equivalence, we need to show that the $n$ conditional statements $p_1 \rightarrow p_2$, $p_2 \rightarrow p_3, \ldots, p_n \rightarrow p_1$ are all true, that is, we have the following:
$$\left(p_1 \leftrightarrow p_2 \leftrightarrow \ldots \leftrightarrow p_n\right) \leftrightarrow\left(\left(p_1 \rightarrow p_2\right) \wedge\left(p_2 \rightarrow p_3\right) \wedge \ldots \wedge\left(p_n \rightarrow p_1\right)\right) .$$

In other words, instead of biconditional statements (if and only if statements), implications (if, then statements) are employed.
Example $4.2$
Prove that for every positive integer $n, n$ is even if and only if $n-1$ is odd.
Solution
There are two statements $p_1: n$ is even, and $p_2: n-1$ is odd. To show they are equivalent, that is, $p_1 \leftrightarrow p_2$, we need to prove that $p_1 \rightarrow p_2$ and $p_2 \rightarrow p_1$. We first prove that if $n$ is even, then $n-1$ is odd. If $n$ is even (i.e., $n$ is an integer that is a multiple of 2 ), then $n=2 k$ for some positive integer $k$, and thus $n-1=2 k-1$, which is odd as it is 1 less than the even number $2 k$. We then prove that if $n-1$ is odd, then $n$ is even. If $n-1$ is odd, then $n-1=2 k+$ 1 for some positive integer $k$, and thus $n=2 k+2=2(k+1)$, which is even as it is a multiple of 2 .

## 数学代写|离散数学代写Discrete Mathematics代考|Proofs of Equivalence

$$\left(p_1 \leftrightarrow p_2 \leftrightarrow \ldots \leftrightarrow p_n\right) \leftrightarrow\left(\left(p_1 \rightarrow p_2\right) \wedge\left(p_2 \rightarrow p_3\right) \wedge \ldots \wedge\left(p_n \rightarrow p_1\right)\right) .$$

$n-1=2 k+1$ 对于一些正整数 $k$ ，因此 $n=2 k+2=2(k+1)$ ，它是 2 的倍数。

## 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。