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# 数学代写|组合学代写Combinatorics代考|MATH233

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## 数学代写|组合学代写Combinatorics代考|Systems of Distinct Representatives

Let $Y$ be a finite set, and let $\mathcal{A}=\left(A_1, A_2, \ldots, A_n\right)$ be a family ${ }^6$ of $n$ subsets of $Y$. A family $\left(e_1, e_2, \ldots, e_n\right)$ of elements of $Y$ is called a system of representatives of $\mathcal{A}$, provided that
$e_1$ is in $A_1, e_2$ is in $A_2, \ldots, e_n$ is in $A_n$.
In a system of representatives, the element $e_i$ belongs to $A_i$ and thus “represents” the set $A_i$. If, in a system of representatives, the elements $e_1, e_2, \ldots, e_n$ are all different, then $\left(e_1, e_2, \ldots, e_n\right)$ is called a system of distinct representatives, abbreviated SDR.

Example. Let $\left(A_1, A_2, A_3, A_4\right)$ be the family of subsets of the set $Y={a, b, c, d, e}$, defined by
$$A_1={a, b, c}, A_2={b, d}, A_3={a, b, d}, A_4={b, d} .$$
Then $(a, b, b, d)$ is a system of representatives, and $(c, b, a, d)$ is an SDR.
A family $\mathcal{A}=\left(A_1, A_2, \ldots, A_n\right)$ of nonempty sets always has a system of representatives. We need pick only one element from each of the sets to obtain a system of representatives. However, the family $\mathcal{A}$ need not have an SDR even though all the sets in the family are nonempty. For instance, if there are two sets in the family, say, $A_1$ and $A_2$, each containing only one element, and the element in $A_1$ is the same as the element in $\mathrm{A}_2$, that is,
$$A_1={x}, A_2={x}$$
then the family $A$ does not have an SDR. This is because, in any system of representatives, $x$ has to represent both $A_1$ and $A_2$, and thus no SDR exists (no matter what $A_3, \ldots, A_n$ equal). However, a family $A$ can fail to have an SDR for somewhat more complicated reasons.

## 数学代写|组合学代写Combinatorics代考|Stable Marriages

In this section ${ }^8$ we consider a variation of the marriage problem discussed in the previous section.

There are $n$ women and $n$ men in a community. Each woman ranks each man in accordance with her preference for that man as a spouse. No ties are allowed, so that if a woman is indifferent between two men, we nonetheless require that she express some preference. The preferences are to be purely ordinal, and thus each woman ranks the men in the order $1,2, \ldots, n$. Similarly, each man ranks the women in the order $1,2, \ldots, n$. There are $n$ ! ways in which the women and men can be paired so that a complete marriage takes place. We say that a complete marriage is unstable, provided that there exist two women $A$ and $B$ and two men $a$ and $b$ such that
(i) $A$ and $a$ get married;
(ii) $B$ and $b$ get married;
(iii) $A$ prefers (i.e., ranks higher) $b$ to $a$;
(iv) $b$ prefers $A$ to $B$.
Thus, in an unstable complete marriage, $A$ and $b$ could act independently of the others and run off with each other, since both would regard their new partner as more preferable than their current spouse. Thus, the complete marriage is “unstable” in the sense that it can be upset by a man and a woman acting together in a manner that is beneficial to both. A complete marriage is called stable, provided it is not unstable. The question that arises first is Does there always exist a stable, complete marriage?

We set up a mathematical model for this problem by using a bipartite graph again. Let $G=(X, \Delta, Y)$ be a bipartite graph in which
$$X=\left{w_1, w_2, \ldots, w_n\right}$$
is the set of $n$ women and
$$Y=\left{m_1, m_2, \ldots, m_n\right}$$is the set of $n$ men. We join each woman-vertex (left is now woman) to each man-vertex (right is now man). The resulting bipartite graph is complete in the sense that it contains all possible edges between its two sets of vertices. ${ }^9$ Corresponding to each edge $\left{w_i, m_j\right}$, there is a pair $p, q$ of numbers where $p$ denotes the position of $m_j$ in $w_i$ ‘s ranking of the men, and $q$ denotes the position of $w_i$ in $m_j$ ‘s ranking of the women. A complete marriage of the women and men corresponds to a perfect matching (of $n$ edges) in this bipartite graph $G$.

## 数学代写|组合学代写Combinatorics代考|Systems of Distinct Representatives

$e_1$在$A_1, e_2$在$A_2, \ldots, e_n$在$A_n$。

$$A_1={a, b, c}, A_2={b, d}, A_3={a, b, d}, A_4={b, d} .$$

$$A_1={x}, A_2={x}$$

## 数学代写|组合学代写Combinatorics代考|Stable Marriages

(一)$A$和$a$结婚;
(二)$B$和$b$结婚;
(iii)相对于$a$, $A$更倾向于(即排名更高)$b$;
(iv) $b$更喜欢$A$而不是$B$。

$$X=\left{w_1, w_2, \ldots, w_n\right}$$

$$Y=\left{m_1, m_2, \ldots, m_n\right}$$的集合是 $n$ 男人。我们将每个女人顶点(左边现在是女人)连接到每个男人顶点(右边现在是男人)。所得到的二部图是完备的，因为它包含了两组顶点之间所有可能的边。 ${ }^9$ 对应于每条边 $\left{w_i, m_j\right}$，有一对 $p, q$ 其中的数 $p$ 表示的位置 $m_j$ 在 $w_i$ 的排名，和 $q$ 表示的位置 $w_i$ 在 $m_j$ 这是对女性的排名。一个完整的男女婚姻对应于一个完美的匹配 $n$ 边)在这个二部图中 $G$．

## 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。