# 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|Multisets

As defined earlier, a set is an unordered collection of objects, where the multiplicity of objects is ignored, and the membership of an object has a binary status, that is, either an element belongs to the set or it does not. We now deviate from this general definition of a set to briefly introduce multisets, where the multiplicity of an object is explicitly significant, and later present fuzzy sets, where membership of an object is not binary but a continuum of values.

A multiset (short form for multiple-membership set), also known as a bag, is an unordered collection of objects where an object can occur as a member of a set more than once, that is, repeated occurrences of objects are allowed. For instance, multisets ${7,8,9}$ and ${9,8,7}$ are the same, but multisets ${7,8,9}$ and ${7,8,7,9}$ are different. The number of occurrences, given for each element, is called the multiplicity of the element in the multiset. A multiset corresponds to an ordinary set if the multiplicity of every element is one.

Example of multisets may include the multiset of prime factors of an integer, such as the integer 360 that has the prime factorization $360=2^3 \times 3^2 \times 5^1$, which gives the multiset ${2,2,2,3,3,5}$. The sets of distinct letters forming the words “are,” “era,” “ear,” and “rear” are the same, which is ${r, a, e}$; however, their multisets of letters forming these words are different, as the multiset of the words “are,” “era,” and “ear” is ${r, a, e}$, whereas that for the word “rear” is ${r, r, a, e}$.

The multiset $A$ is a subbag of the multiset $B$, that is, $A \subseteq B$, if the number of occurrences of each element $x$ in $A$ is less than or equal to the number of occurrences of $x$ in $B$. For instance, if $A={a, b, c, b}$ and $B={a, b, c, a, b}$, then $A$ is a subbag of $B$, but $B$ is not a subbag of $A$. Two bags $A$ and $B$ are equal if and only if $A$ is a subbag of $B$ and $B$ is a subbag of $A$.

The notation $\left{m_1 \cdot a_1, m_2 \cdot a_2, \ldots, m_n \cdot a_n\right}$ denotes the multiset with the element $a_1$ occurring $m_1$ times, the element $a_2$ occurring $m_2$ times, and so on. The numbers $m_i, i=1, \ldots, n$ are called the multiplicities of the elements $a_i, i=1, \ldots, n$, where elements not in a multiset are assigned 0 as their multiplicity. The cardinality of a multiset is determined by summing up the multiplicities of all its elements, that is, $m_1+m_2+\ldots+m_n$. For example, in the multiset ${c, a, n, a, d, i, a, n}$, the multiplicities of the distinct members $c, a, n, d$, and $i$ are respectively $1,3,2,1$, and 1 , and therefore the cardinality of this multiset is $8(=1+3+2+1+1)$.

The union or intersection of two multisets is the multiset in which the multiplicity of an element is the maximum or the minimum of its multiplicities in those two multisets, respectively. The difference of two multisets is the multiset in which the multiplicity of an element is the difference between the multiplicities of the element in these two multisets, unless the difference is negative, in which case the multiplicity is 0 . The sum of two multisets is the multiset in which the multiplicity of an element is the sum of multiplicities in those two multisets.

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

In a world of many shades of gray, a black-white dichotomy is an unnecessary artificial imposition. The concept of fuzzy sets is an important and practical generalization of the notion of classical sets. For instance, if the universe of discourse consists of knowledgeable people, then in fuzzy set theory, members of a set can have varying degrees of knowledge. Fuzzy sets, introduced by Lotfi Zadeh, where each member of the set is defined by the degree of fuzziness, have an array of applications in modeling, control systems, linguistics, information retrieval, decision-making, and of course artificial intelligence, where information is incomplete or imprecise.

In classical set theory, a set $A$ is defined in terms of its characteristic function $\mu_A(x)$, a mapping from the universal set $U$ to the binary set ${0,1}$, where $x$ belongs to $A$ if and only if $\mu_A(x)=1$ and $x$ does not belong to $A$ if and only if $\mu_A(x)=0$. In fuzzy set theory, a set $A$ is defined in terms of its membership function $\mu_A(x)$, a mapping from the universal set $U$ to the unit interval $[0,1]$, where $x$ in the fuzzy set $A$ has a certain degree of membership. Therefore the fuzzy set $A$ is denoted by listing the elements with their degrees of membership.

Classical sets are special cases of fuzzy sets, in which the membership functions of fuzzy sets only take values 0 or 1 . In the context of fuzzy sets, classical sets are usually called crisp sets. For instance, the membership functions for fuzzy and crisp sets of tall people reflecting their degrees of tallness are shown in Fig. 5.5. The crisp set assigns a number from the binary set ${0,1}$ to indicate whether a person is considered tall or not (e.g., whether the person’s height is greater than or less than $180 \mathrm{~cm}$ ), whereas the fuzzy set assigns a real number in the interval $[0,1]$ to indicate the extent to which a person is a member of the set of tall people (e.g., the person’s height ranges between $170 \mathrm{~cm}$ and $190 \mathrm{~cm}$ ).
The degree of fuzziness for each member of the fuzzy set needs to be always specifically stated, noting that elements with 0 degree of membership are not listed. As an example, the fuzzy set $A$ of healthy people consists of $a, b, c, d$, and $e$, whose degrees of membership (i.e., degrees of healthiness) are as follows: $\mu_A(a)=0.99, \mu_A(b)=0.9$, $\mu_A(c)=0.5, \mu_A(d)=0.05$, and $\mu_A(e)=0.001$. In turn, this points to $a$ being the healthiest and $e$ having the poorest health in the fuzzy set $A$. As another example, the fuzzy set $B$ of wealthy people consists of $a, b, c$, and $d$, whose degrees of membership (i.e., degrees of wealthiness) are as follows: $\mu_B(a)=0.999, \mu_B(b)=0.95, \mu_B(c)=0.2$, and $\mu_B(d)=0.001$. This in turn indicates that $a$ is the wealthiest and $d$ is the poorest in the fuzzy set $B$.

The concepts of set inclusion and equality can also be extended to fuzzy sets. Assuming $A$ and $B$ are fuzzy sets, we have $A \subset B$, that is, $A$ is a proper subset of $B$, if and only if for every element $x$, we have $\mu_A(x)<\mu_B(x)$, and we have $A=B$ if and only if for every element $x$, we have $\mu_A(x)=\mu_B(x)$.

Set operations in classical sets can be extended to fuzzy sets in terms of membership function, namely, we have

• The complement of fuzzy set $A$ is $A^c$, where $\mu_A(x)=1-\mu_A(x)$.
• The union of fuzzy sets $A$ and $B$ is $A \cup B$, where $\mu_{A \cup B}(x)=\max \left{\mu_A(x), \mu_B(x)\right}$.
• The intersection of fuzzy sets $A$ and $B$ is $A \cap B$, where $\mu_{A \cap B}(x)=$ $\min \left{\mu_A(x), \mu_B(x)\right}$

# 离散数学代写

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

Veft{m_1 \cdot a_1, m_2 \cdot a_2, \Idots, $m_{-} n \backslash c d o t$ a_n $n i$

$a_i, i=1, \ldots, n$ ，其中不在多重集中的元素被分配 0

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

$\mu_B(a)=0.999, \mu_B(b)=0.95, \mu_B(c)=0.2$ ， 和 $\mu_B(d)=0.001$. 这反过来说明 $a$ 是最富有的 $d$ 是模糊集 中最差的 $B$.

$\mu_A(x)<\mu_B(x)$ ，我们有 $A=B$ 当且仅当对于每个元 素 $x$ ，我们有 $\mu_A(x)=\mu_B(x)$.

• 模糊集的补集 $A$ 是 $A^c$ ，在哪里 $\mu_A(x)=1-\mu_A(x)$
• 模糊集的并集 $A$ 和 $B$ 是 $A \cup B$ ，在哪里
• 模糊集的交集 $A$ 和 $B$ 是 $A \cap B$ ，在哪里 $\mu_{A \cap B}(x)=\backslash \min \backslash$ left ${$ mu_A $\mathrm{A}(\mathrm{x}), \backslash$ mu_B $\mathrm{B}(\mathrm{x}) \backslash$ right $}$

## 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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