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数学代写|离散数学代写Discrete Mathematics代考|MTH645 Basic Definitions

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数学代写|离散数学代写Discrete Mathematics代考|Basic Definitions

A Boolean algebra is a mathematical system. A Boolean algebra consists of a nonempty set $B$ together with two binary operations of addition ” + ” and multiplication “.”, where they map elements of $B \times B$ to elements of $B$ (i.e., if $x, y \in B$, then $x+y$ and $x \cdot y$ are also in $B$ ), a unary operation of complementation “،,” where it maps elements of $B$ to elements of $B$, two distinct elements ” 1 ” and ” 0 “, and the axioms for all elements $x, y$, and $z$ in $B$, as summarized in Table 8.1 .

In symbols, a Boolean algebra is designated by its six parts $\left{B,+, \cdot,{ }^{\prime}, 0,1\right}$. The operators “+”, “.”, and “،’, are called sum, product, and complement, respectively. Note that “+”” and “.” are not the usual arithmetic operators “plus” and “times.” The symbols ” 0 ” and ” 1 “, which are called the zero element and the unit element, respectively, do not represent numbers on the real number line. However, the names “plus,” “times,” “complement,” “zero,” and “one” are commonly used informally when discussing Boolean algebras. For convenience, ” $x \cdot y$ ” is shown as ” $x y$ “, and it is also common to replace the complement operation “،,” by a bar “”, (e.g., $\left.x^{\prime}=\bar{x}\right)$.

数学代写|离散数学代写Discrete Mathematics代考|Boolean Expressions and Boolean Functions

Let $B={0,1}$. The variable $x$ is called a Boolean variable if it assumes values only from $B$ (i.e., if its only possible values are 0 and 1). Then, $B^n=$ $\left{x_1, x_2, \ldots, x_n \mid x_i \in B\right.$ for $\left.1 \leq i \leq n\right}$ is the set of all possible $n$-tuples of 0 s and $1 \mathrm{~s}$ and has $2^n$ elements. A function from $B^n$ to $B$ is called a Boolean function of degree $\boldsymbol{n}$. Boolean functions can be defined by Boolean tables. Because a Boolean function is an assignment of 0 or 1 to each of these $2^n$ different $n$-tuples, there are $(2)^{\left(2^n\right)}$ different Boolean functions of degree $n$, labeled $F_1, F_2, \ldots, F_{2^2}$. Table 8.2 presents all Boolean functions of degree two (i.e., $n=2$ ), labeled $F_1, F_2, \ldots, F_{16}$.

A Boolean expression consists of Boolean variables and Boolean operators. The Boolean expression in the Boolean variables $x_1, x_2, \ldots, x_n$ are defined recursively through the basic clause that states that $0,1, x_1, x_2, \ldots, x_n$ are Boolean expressions, and the recursive clause that states that the sum and the product of any two Boolean expressions as well as the complement of any Boolean expression are also Boolean expressions. Each Boolean expression represents a Boolean function. The values of a Boolean function are obtained by substituting 0 and 1 for the Boolean variables in the Boolean expression.

数学代写|离散数学代写Discrete Mathematics代考|Boolean Expressions and Boolean Functions

(即，如果它的唯一可能值是 0 和 1)。然后， $B^n=$ 合 $n$-元组 0 和 $1 \mathrm{~s}$ 并且有 $2^n$ 元素。一个函数来自 $B^n$ 到 $B$ 被称为度数的布尔函数 $\boldsymbol{n}$. 布尔函数可以由 布尔表定义。因为布尔函数是对这些中的每一个赋值 0 或 $12^n$ 不同的 $n$-元组，有 $(2)^{\left(2^n\right)}$ 度的不同布 尔函数 $n$ ，标记 $F_1, F_2, \ldots, F_{2^2}$. 表 8.2 列出了所有二阶布尔函数 (即， $n=2$ ), 标记 $F_1, F_2, \ldots, F_{16}$

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。