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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 英国补考|组合学代写Combinatorics代考|The Notion of a Set

A set is a distinct collection of certain things, creatures, symbols, or other objects. The objects that make up a set are called its elements. In order to make a distinction between one set and the others, one needs to know the pattern, which distinguishes the objects of this set from all other objects. It is in this sense that the words “distinct collection” are to be understood. Here are the examples of sets: the set of points of a given segment; set of vertices of a given triangle; set of all natural numbers; set of two-digit positive integers; set of letters of the Ukrainian alphabet; set of all the words used by Taras Shevchenko in the poem “The Caucasus”, etc.

It is usual to denote sets by capital letters of various alphabets and their elements by lower case letters or other symbols. For the most important numeric sets there are fixed notation: $N$ is used to denote the set of all natural numbers, $Z$ is reserved for the set of all integer numbers, $Q$ denotes the set of all rational numbers and $R$ is a conventional notation for the set real numbers. In other cases, the notation is optional and should be clearly introduced before it is used.

If $M$ is a set, then the formalized expression $a \in M$ (which reads ” $a$ is an element of $M$ ” or ” $a$ belongs to $M$ “) means that $a$ is its element. The fact that $a$ is not an element of the set $M$ is expressed by the notation $a \notin M$. For example, the following expressions are correct: $2 \in N, 2 \in R, \sqrt{2} \in R, \sqrt{2} \notin Q, \pi \notin Z$.

The equality sign can be placed between two symbols (letters) denoting sets only if they denote the same set. If $A$ and $B$ are sets and $A=B$, then it actually means that $A$ and $B$ is the same set. For instance, let $A$ be the set of all two-digit natural numbers and $B$ be the set of all natural numbers in the interval $(9,100)$. Then $A=B$ as both sets have the same composition.

Depending on the number of their elements, sets can be finite or infinite. Finite sets are those, the number of elements of which can be expressed with a natural number. In other words, the set $A$ is finite if it is possible to establish a bijection between its elements and the interval of the natural series (from 1 to some number $n$ ). For example, the set of one-digit natural numbers is finite. It is composed of 9 elements (numbers). The set of letters of Ukrainian alphabet is also finite. Other examples of finite sets include: integer solutions to the inequality $|x| \leq 10$; solutions of the equation $x^3-4 x=0$; possible dispositions of pieces on a chessboard, which can evolve during the game of chess. The latter set is incredibly large, but still it is finite. Nobody can count all possible combinations on the chessboard, even the most powerful computer. However, it is possible to find a number, which exceeds the number of such dispositions. We will make it below.

## 英国补考|组合学代写Combinatorics代考|Subsets

If all elements of a set $A$ belong to a set $B$, then $A$ is called a subset of the set $B$. This is expressed as follows: $A \subset B$ (this expression reads ” $A$ is a subset of $B$ “). It is also worth pninting out that for any set $R$ the sets $\emptyset$ and $R$ are subsets of the set $R$. This is provided hy the definition. Really, any element of the set $\emptyset$ belongs to the set $B$, as there are no elements in $\emptyset$. Hence, $\emptyset \subset B$. Also, $B \subset B$. In this case, the condition in the definition turns into tautology. Other subsets of the set $B$ (if any) are non-empty and do not coincide with the set $B$ itself. They could be considered to be true parts of $B$. Such subsets are called proper non-empty subsets of the set $B$. For example, here is the list of all the subsets of the set ${a, b, c}$
$$\text { 0, }{a},{b},{c},{a, b},{a, c},{b, c},{a, b, c} .$$
There are 6 proper non-empty subsets among them (all, except for the first and the last one). Below, we provide several other examples of subsets.

Example 2.4. Let $R$ be the set of all real numbers. It has an infinite amount of subsets, including $Q, Z, N,[-1,1]$ (the set of all numbers in the interval from $-1$ to 1 inclusive),

$(0, \infty)$ (the set of all positive numbers), the set of square roots of all natural numbers, the set of roots of all natural powers of 2 , etc.

Example 2.5. Denote the set of all English words in the Oxford English Dictionary by $C$. Among its subsets there are: words, beginning with a consonant letter; two-syllable words; nouns; verbs; words, beginning with ” $a$ “; words with no closed syllables; words, having vowels at their start and end; words, having double consonant; nouns ending with the letter “o” and many others.

Infinite sets have an infinite amount of subsets. This follows straightforwardly from the fact that they have an infinite amount of singletons, that is, subsets consisting of only one element. On the contrary, finite sets have a finite amount of subsets, which can be divided into groups by the number of their elements. Finding the amount of subsets in a certain group is one of the central combinatorial problems. We will solve it in the next section.
The notion of a set is twofold. From one point of view, a set is a collection of certain objects, which are called its elements. Alternatively, a set itself is an individual object, which in particular, can be an element of other sets. For instance, the group (set) of students of a given school consists of separate individuals and at the same time have patterns of the individual unit itself. Indeed, the groups of students from different schools can interact with each other similarly to separate individuals: organize sports or intellectual competitions, share the information and experience in self-regulation, free time activities, etc. The phrase “London school teams”‘ sounds absolutely correct, despite formally it tells about the set, elements of which are other sets.

Let $A$ be a set. Its subsets can be elements of other sets. In particular, it is possible to create a set consisting of all the subsets of the set $A$ (and only of them). This set is called consistently: the set of all subsets of the set $A$. It is denoted by $S(A)$. For example, if $A={a, b, c}$, then $S(A)={0,{a},{b},{c},{a, b},{a, c},{b, c}, A}$. For any given set $A$, the sets $A$ and $\emptyset$ are two of the elements of $S(A)$. In the case $A=\emptyset$, there is only one element in $S(A)$.

# 组合学代考

## 英国补考|组合学代写Combinatorics代考|Subsets

0, 一个,b,C,一个,b,一个,C,b,C,一个,b,C.

(0,∞)（所有正数的集合），所有自然数的平方根的集合，所有 2 的自然幂的根的集合，等等。

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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。

assignmentutor™您的专属作业导师
assignmentutor™您的专属作业导师