## 数学代写|离散数学作业代写discrete mathematics代考|MPCS50103

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## 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, $A \cap A^c=\varnothing$. 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
$$A \oplus B=A \Delta B \triangleq{x \in U \mid(x \in A, x \notin B) \text { or }(x \notin A, x \in B)}$$

## 数学代写|离散数学作业代写discrete mathematics代考|Set Identities and Methods of Proof

A set identity is an equality between two set expressions that is true for all elements of the sets involved in the identity. In a set identity, some basic set operations are combined to form another set. Table 5.2 presents some important set identities. These identities in set theory are similar to the logical equivalences in logic.

It is a fact of set algebra, called the principle of duality, that the dual of an identity is also an identity, where the dual of an identity can be obtained by replacing each occurrence of $\cup, \cap, U$, and $\varnothing$ in the identity by $\cap, \cup, \varnothing$, and $U$, respectively. Many of the identities in Table 5.2 arranged in pairs reflect the principle of duality.

De Morgan’s laws are prominent set identities that provide a pair of transformation rules. The laws can be expressed as the complement of the union of two sets is the same as the intersection of their complements, and the complement of the intersection of two sets is the same as the union of their complements. De Morgan’s laws are used when the complements of sets are easier to define than the sets themselves.

To prove set identities, membership tables can be used. A table that displays the membership of elements in sets is called a membership table, also known as a truth table. The columns of a membership table must represent the original basic sets and the two sets on both sides of the set identity, where 1 is used to indicate an element that is in the set, and 0 is used to indicate an element that is not in the set. Note that there is a great similarity between membership tables in set theory and truth tables in propositional logic.

Table 5.3 presents a membership table for the union, intersection, difference, and symmetric difference of two sets, as well as the complements of the two sets.

Insights regarding set identities can be obtained from Venn diagrams, but Venn diagrams cannot be used when proving theorems unless special attention is paid to make sure that the diagrams are sufficiently general to encompass all possible cases, and that is a difficult task. As the role of Venn diagrams is not to provide formal proofs, we need formal methods of proving set identities. Here are three distinct methods to prove a set identity:

Show each side of the identity is a subset of the other side. This method of proof is known as the element argument or containment proof. In other words, to prove $M=K$, we need to prove $M \subseteq K$ and $K \subseteq M$. This powerful method brings insight into the proof, but in some cases, this proof method may prove to be rather complex.

Transform one side into the other side step by step by employing the other known set identities. This method of proof is known as the algebraic proof. This is usually the shortest method, provided that there are relevant set identities that can be applied to simplify the set expressions.

Build a membership table step by step for each side of the set identity, and show the columns corresponding to the both sides of the identity are identical. This method, known as proof by membership table, does not provide any insight into the proof. However, it is a straightforward method if the number of the original sets in the identity is just a few, otherwise, a computer should be used to build the membership table of interest.

# 离散数学代写

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

$$A \cup B \triangleq x \in U \mid x \in A \text { or } x \in B$$

$A \cap B \triangleq x \in U \mid x \in A$ and $x \in B$

$$A-B \triangleq x \in U \mid x \in A \text { and } x \notin B$$

$$A^c=\bar{A} \triangleq x \in U \mid x \notin A$$

$$A \oplus B=A \Delta B \triangleq x \in U \mid(x \in A, x \notin B) \text { or }(x \notin A$$

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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。