数学代写|离散数学作业代写discrete mathematics代考|MATH200

2023年3月31日

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数学代写|离散数学作业代写discrete mathematics代考|Definitions and Notation

A set is an unordered collection of distinct objects that are called elements or members of the set. It is essential to have a clear and rigorous definition of a set. For instance, “smart children in a town” does not form a set, as the word “smart” does not have a universally agreeable definition, and its membership is debatable, whereas “pregnant women in a town” does form a well-defined set.

It is common to use capital letters, such as $A$, to denote sets, and lowercase letters, such as $x$, to refer to set elements. If $x$ is an element of the set $A$ or equivalently $x$ belongs to $A$, we then use the notation $x \in A$, and if $x$ does not belong to the set $A$ or equivalently $x$ is not an element of $A$, we then write $x \notin A$. For instance, if $A$ is the set of all capital cities, then Tokyo, denoted by $x$, is an element of $A$, that is, $x \in A$, and if $B$ is the set of all European cities, then Tokyo, denoted by $x$, is not a member of $B$, because it is a city in Asia, we thus have $x \notin B$.

A set is generally represented by braces (curly brackets), that is, by {} . One way to specify a set with a finite number of elements is to use the set roster method, by which all the elements of the set are listed between curly brackets (i.e., within braces), such as ${3,6,9}$. The order of elements presented in a set is irrelevant, and a set remains the same if its elements are repeated or rearranged. Note that a set of a very large number of elements that follow a recognizable pattern is usually described by listing the first few elements, followed by ellipses “…,” which is read as “and so forth,” such as ${1,2,3,4, \ldots}$

Another way to specify a set is the set builder notation, through which some property held only by all members of the set is clearly and completely described, such as ${x \in N \mid x$ is a multiple of $3,0<x<10}$, where the vertical line ( $\mid$ ) is read as “such that” and the comma $($,$) as “and,” and \boldsymbol{N}$ represents the set of all positive integers. Note that the general form ${x \in S \mid Q(x)}$, where $Q(x)$ is a predicate indicating the property that the object $x$ of the set $S$ has, is read as “the set of all $x$ in $S$ such that $x$ has the property $Q(x) . “$

A set usually presents a group of elements with common properties. However, it is possible for a set to contain any kind of elements whatsoever, and they are not required to be of the same type, such as the set {China, nose, baby, movie, ice cream, $\pi$, rainbow, stamp, soccer $}$.

A Venn diagram is a group of simple closed curves arranged in a plane to visually illustrate collections of sets and their logical relationships through geometric intuition so as to help understand set concepts and operations. Fig. 5.1 shows the Venn diagrams for some special sets.

数学代写|离散数学作业代写discrete mathematics代考|Set Operations

As propositions can be combined to construct new propositions in various ways, sets can be combined to build a new set, which then has a certain property. There is a close relationship between logic operations and set operations. Fig. 5.3 shows the Venn diagrams for some special set operations.

The union of two sets $A$ and $B$, denoted by $A \cup B$, is the set of all elements that are in $A$ or in $B$ or in both, as shown in Fig. 5.3a, that is, we have
$$A \cup B \triangleq{x \in U \mid x \in A \text { or } x \in B}$$
Here, “or” within the curly brackets is used in the sense of “and” as well as “or”, thus it implies at least in one of the two sets. The intersection of two sets $A$ and $B$, denoted by $A \cap B$, is the set of all elements that exist in both $A$ and $B$, as shown in Fig. 5.3b, that is, we have
$$A \cap B \triangleq{x \in U \mid x \in A \text { and } x \in B} .$$
The intersection of two disjoint sets $A$ and $B$ is thus the empty set, that is, $A \cap B=\varnothing$. The difference of sets $A$ and $B$ (or the relative complement of $B$ with respect to $A$ ), denoted by $A-B$ or $A \backslash B$, is the set of elements in $A$ that are not in $B$, as shown in Fig. 5.3c, that is, we have
$$A-B \triangleq{x \in U \mid x \in A \text { and } x \notin B}$$
Note that the set $A-B$, read as ” $A$ minus $B$,” is different from the set $B-A$. The $\boldsymbol{a} b$ solute complement or, simply, the complement of a set $A$, with respect to the universal set $U$, denoted by $A^c$ or $\bar{A}$, is the set of all elements that are not in $A$, as shown in Fig. 5.3d, that is, we have
$$A^c=\bar{A} \triangleq{x \in U \mid x \notin A}$$
Note that the complement of the universal set is the empty set and vice versa, the union of a set and its complement is the universal set, that is, $A \cup A^c=U$, and the intersection of a set and its complement is the empty set, that is, $A \cap A^{\varepsilon}=\varnothing$. The symmetric difference of sets $A$ and $B$, denoted by $A \oplus B$ or $A \Delta B$, consists of those elements that belong to $A$ or $B$ but not to both, as shown in Fig. 5.3e, that is, we have
$$A \oplus B=A \Delta B \triangleq{x \in U \mid(x \in A, x \notin B) \text { or }(x \notin A, x \in B)} .$$
We thus have
$$A \oplus B=(A \cup B)-(A \cap B)$$

离散数学代写

有限元方法代写

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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。