## 数学代写|离散数学作业代写discrete mathematics代考|MATH 300

2022年7月12日

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## 数学代写|离散数学作业代写discrete mathematics代考|Algebraic Theory

In Chapter 2, we studied propositions, that is, logical constants, which are either true or false statements, but not both. For instance, $5=4$ and $5=$ $2+3$ are examples of a false and of a true proposition, respectively. However, there are many likewise sentences, which are not propositions. For instance, we cannot determine the truth value of a sentence $5=x+3$ if we do not know a value of $x$; therefore, this sentence is not a proposition. Since similar examples penetrate mathematics and sciences, it is necessary to include them in our theory. To give a definition, let us note that the linear equation $5=x+3$ is a propositional function with the argument $x$, whose domain may be the set of integers, and the range of this function is the two-element set {true, false}. Such maps are called predicates. We develop here a purely algebraic non-formal, non-axiomatic theory parallel and in a sense equivalent to the theory of relations in $n$ indeterminates.

Definition 23. A Boolean predicate in one unknown or one indeterminate $x$ (unary predicate) is any function $P(x): D \rightarrow B$ with the domain $D$ and the range $B={$ true, false $}$; this range is equivalent to the two-element set $B_{2}={1,0}$.

Example 12. For example, let $D$ be the set $\mathcal{R}$ of the real numbers and the predicate
$P(x): x^{2}$ is positive.

## 数学代写|离散数学作业代写discrete mathematics代考|QUANTIFIERS

However, there are two other operations with the predicates that have no similar operations with propositions-these are quantifiers, the existential quantifier and the universal quantifier. Consider the phrase “There exists an element $x$ such that the statement $P(x)$ is valid.” If $P(x)$ is identically false, then that statement is false; otherwise, it is true. Hence we have a map from one-variable predicates into the simplest Boolean algebra $B_{2}$, and we can straightforwardly translate it into the language of sets and maps.

Definition 26. Consider a family of unary predicates ${P}$ with the fixed domain D. (1) The map $\exists P$, denoted also as $\exists x(P(x))$, from that family to $B_{2}$, such that $\exists P \equiv 0$ if $P$ is identically false and $\exists P \equiv 1$ otherwise, is called the existential quantifier (on that family).
(2) The map $\forall P$, denoted also as $\forall x(P(x))$, from that family to $B_{\mathfrak{s}}$, such that $\forall P \equiv 0$ if $P$ is not identically true and $\forall P \equiv 1$ if $P$ is identically true, is called the universal quantifier (on that family).

For instance, the statement “There exists an integer divisor of 25 ” is true, the statement “There exists an integer divisor of $29 “$ is false, and the statement “There are infinitely many prime numbers-twins” is not a proposition; as of now (May of 2021) we don’t know whether it is true or false since the twin-primes hypothesis has not been proved or disproved yet. On the other hand, the statement “Every integer divides 25 ” is false, while the statement “The square of every real number is non-negative” is true.

## 数学代写|离散数学作业代写discrete mathematics代考| QUANTIFIERS

(2) 地图 $\forall P$ ，也表示为 $\forall x(P(x))$ ，从那个家庭到 $B_{5}$ ，使得 $\forall P \equiv 0$ 如果 $P$ 并不完全 相同，并且 $\forall P \equiv 1$ 如果 $P$ 同样是正确的，称为通用量词 (在该族上)。

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## 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。