# 数学代写|离散数学作业代写discrete mathematics代考|CS3653

#### Doug I. Jones

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## 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:

1. 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.
2. 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.
3. 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.

## 数学代写|离散数学作业代写discrete mathematics代考|Cardinality of Sets

The number of distinct elements in a set $A$ is called the cardinality of $A$, written as $|A|$. The cardinality of a set (i.e., the size of a set) may be finite or infinite. For instance, we have $|\varnothing|=0$ because the empty set has no elements. A set with a finite number of elements is defined as a finite set, and it is thus countable. The exact number of elements in a finite set can be known, such as the set of cards in a deck of playing cards, or unknown,

such as the set of fish in the world. A set that is not finite is infinite, an infinite set is either countable or uncountable. In a countably infinite set, it is possible to list the elements of the set in a sequence indexed by positive integers, such as the set of all prime numbers. On the other hand, in an uncountably infinite set, it is not possible to list the elements of the set in a sequence indexed by positive integers, such as the set of all real numbers between 0 and 1.

A set can have other sets as members. The set of all subsets of a set $A$, which also includes the empty set $\varnothing$ and the set $A$ itself, is called the power set of $A$ and is denoted by $P(A)$. If $A$ is a finite set, then we have
$$|P(A)|=2^{|A|}$$
which in turn implies $|A|<|P(A)|$. Given two sets of $A$ and $B$, the Cartesian product of $A$ and $B$, denoted by $A \times B$ and read as ” $A$ cross $B$,” is the set of all ordered pairs $(a, b)$, where $a \in A$ and $b \in B$. The number of ordered pairs in the Cartesian product of $A$ and $B$ is equal to the product of the number of elements in the set $A$ and the number of elements in the set $B$, that is, $|A \times B|=|A||B|$. The Cartesian product of more than two sets can also be defined. The Cartesian product of $n$ sets $A_1, A_2, \ldots, A_n$ is the set of all ordered $\boldsymbol{n}$-tuples, and symbolically is shown as follows:
$$A_1 \times A_2 \times \ldots \times A_n=\left{\left(a_1, a_2, \ldots, a_n\right) \mid a_1 \in A_1, a_2 \in A_2, \ldots, a_n \in A_n\right} .$$
The notation for an ordered $n$-tuple is a generalization of the notation for an ordered pair, and it takes both order and multiplicity into account.

A subset $R$ of the Cartesian product $A \times B$ is called a relation from the set $A$ to the set $B$. The elements of $R$ are ordered pairs, where the first element belongs to $A$ and the second to $B$. In general, we have $A \times B \neq B \times A$, unless $A=\varnothing, B=\varnothing$, or $A=B$.

# 离散数学代写

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

1. 表明身份的每一面都是另一面的子集。这种证明方法称为元素论证或包含证明。换句话说，要证明米=钾, 我们需要证明米⊆钾和钾⊆米. 这种强大的方法带来了对证明的洞察力，但在某些情况下，这种证明方法可能被证明是相当复杂的。
2. 通过使用其他已知的集合标识，逐步将一侧转换为另一侧。这种证明方法称为代数证明。这通常是最短的方法，前提是存在可用于简化集合表达式的相关集合恒等式。
3. 对集合身份的每一边逐级建立隶属度表，并显示身份两边对应的列是相同的。这种方法称为成员资格表证明，不提供对证明的任何洞察。但是，如果恒等式中的原始集的数量只有几个，这是一种直接的方法，否则，应该使用计算机来构建感兴趣的隶属度表。

## 数学代写|离散数学作业代写discrete mathematics代考|Cardinality of Sets

$$|P(A)|=2^{|A|}$$

$A_1, A_2, \ldots, A_n$ 是所有有序的集合 $\boldsymbol{n}$-元组，并且符号 化地显示如下:

## 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。

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