## 数学代写|离散数学作业代写discrete mathematics代考|MATH-UA 120

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|Commutativity of Quantifiers and Boolean Operations

We know that $2+3 \times 4=14 \neq 2 \times 3+4=10$. Since the Boolean operations with predicates are defined through those operations with propositions, the Boolean operations also are not always commute with one another, for instance, $p \vee q \wedge r \not \neq p q \vee r$, however, this is important, and we have to study how these operations “communicate” with each other. The definitions must provide for the uniqueness of the reading of every formula. To achieve that, we are to define the order of operations. Moreover, we often have to use separators-parentheses, etc. Since we work with logical objects, with the predicates and propositions, we often have to write that these objects are equivalent, that is, $p \equiv q$, etc. However, we treat them as algebraic objects; therefore, we sometimes write $p=q$ with the same meaning.

An important conclusion is that $p \vee q \wedge r \not \equiv p \wedge q \vee r$; and moreover, in general $\forall(P \vee Q) \not(\forall P) \vee(\forall Q)$. To derive valid formulas, first of all we consider a special case, when the domains of the predicates under consideration are finite sets. In the case of infinite families, the domains considered must be proper sets and not classes.

Example 17. Consider a predicate ${P(x): x>0, x \in Z}$, that is, the property of the integers to be positive. The domain of the predicate is the set of integers $\mathcal{Z}$ and the truth domain is ${1,2,3, \ldots}$, thus this predicate is not identically true nor identically false, it is satisfiable.

Example 18. Next, we consider a predicate $\left{P_{9}(x): 0 \leq x \leq 9, x \in \mathcal{Z}\right}$, that is, property of the integers between 0 and 9 inclusive to be positive. The domain of the predicate is $\mathcal{Z}$, and the truth domain is ${1,2, \ldots, 8,9}$; thus this predicate also is satisfiable but not identically true.

The truth values of this predicate can be easily calculated in finitely many steps without any regard to the quantifiers since the predicate can be written as a finite conjunction
$$P_{9} \equiv(0>0) \wedge(1>0) \wedge(2>0) \wedge \cdots \wedge(9>0),$$ and clearly, $\forall x\left(P_{9}(x)\right) \equiv$ False, since $P(0) \equiv 0$; in other words, $P_{9}$ is not an identically true predicate.

## 数学代写|离散数学作业代写discrete mathematics代考|ORDERED TOTALITIES AND CARTESIAN PRODUCTS

Consider a two-element set ${a, b}$ – the braces here indicate that this is a set without any ordering of its elements, thus, ${a, b}={b, a}$, the latter is the same set, whether $a=b$ or $a \neq b$. However, in many problems, we have to distinguish ordered totalities. Thus, the car license plates $A-123$ and $A-312$ are two different plates. We emphasize that the sets are unordered totalities, denoted by (curly) braces, while the ordered totalities are denoted by parentheses. It is possible to define the ordered totalities through the maps, but we do not go into these details and treat ordered totalities as primary concepts. The major feature, distinguishing the ordered pairs from non-ordered sets is that $(a, b) \neq(b, a)$ as long as $a \neq b$, while ${a, b}={b, a}$ always.

As another example, let a family have two sons, John and Peter. If we consider a set of these sons, then two writings, ${$ John, Peter $}$ and ${$ Peter, John $}$ represent the same set of children, ${$ John, Peter $}={$ Peter, John $}$. On the other hand, if these two words represent the first name and the second name of an individual, then definitely Mr. John Peter Doe and Mr. Peter John Doe are two different persons. To work with ordered totalities, we need new concepts and definitions.

Likewise, we can define new objects, such that an ordered triple $(a, b, c)$ with the first element $a$, the second element $b$, and the third element $c$, an ordered quadruple $(a, b, c, d), \ldots$, an ordered $n$-tuple $\left(a_{1}, a_{2}, a_{3}, \ldots, a_{n}\right)$, etc. For example, for any person, her first, second, and last names make an ordered triple, such as Mary Patty Doe. Social security numbers are ordered 9-tuples; we must regard 123-45-6789 and 213-45-6789 as distinct SS-numbers, even though the numerals in both numbers form the same set ${1,2,3,4,5,6,7,8,9}$. The same is true for telephone numbers: it is safe to call (718) $900-0000$, but it may cost you a lot of money to dial (900) $718-0000$.

## 数学代写|离散数学作业代写discrete mathematics代考| Commutativity of Quantifiers and Boolean Operations

$$P_{9} \equiv(0>0) \wedge(1>0) \wedge(2>0) \wedge \cdots \wedge(9>0),$$

## 数学代写|离散数学作业代写discrete mathematics代考| ORDERED TOTALITIES AND CARTESIAN PRODUCTS

$a, b=b, a$ ，后者是同一夽，无论 $a=b$ 或 $a \neq b$. 但是，在许多问题中，我们必须区分有 序的整体性。因此，汽车牌照 $A-123$ 和 $A-312$ 是两个不同的板块。我们强调堆合是 无序的总计，由 (大括昊) 表示，而有序的总计由括昊表示。可以通过映射定义有序的总 计，但我们不会深入到这些细节中，也不会将有序的总计视为主要概念。区忩有序对和非 有序焦的主要特征是 $(a, b) \neq(b, a)$ 只要 $a \neq b$ 而 $a, b=b, a$ 总是。

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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。