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# 数学代写|离散数学代写Discrete Mathematics代考|MTH645 Multisets

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## 数学代写|离散数学代写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 ${r, r, a, e}$.

## 数学代写|离散数学代写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.

## 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。