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# 数学代写|组合学代写Combinatorics代考|The Pigeonhole Principle

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## 数学代写|组合学代写Combinatorics代考|The Pigeonhole Principle

The Pigeonhole Principle is a method for obtaining statements of the form
If $n \geq g(d)$, then $\mathcal{A}(n, d)$,
where $\mathcal{A}$ is a statement and $g$ is some function depending on $\mathcal{A}$. For example, we’ll see that, if $n \geq d^2+1$, then any sequence of $n$ distinct numbers contains a monotonic subsequence of length $d$. (What follows “then” is $\mathcal{A}(n, d)$.)
Here is a statement of the principle in two forms.
Theorem 2.5 Pigeonhole Principle Function form: Suppose $A$ and $B$ are sets with $|A|>|B|$, then for every function $f: A \rightarrow B$ there is a $b \in B$ with $\left|f^{-1}(b)\right|>1$.
Partition form: Suppose $\mathcal{P}$ is a partition of the set $A$ into less than $|A|$ blocks. Then some block contains more than one element of $A$.

You should be able to prove this theorem. In fact, it is so simple we should probably not even call it a theorem. To see why the two forms of the theorem are equivalent, first suppose $f: A \rightarrow B$. Let $\mathcal{P}$ be the coimage of $f$. It must have at most $|B|<|A|$ blocks. Conversely suppose $\mathcal{P}$ is a partition of $A$. Number the blocks in some fashion from 1 to $|\mathcal{P}|$. Let $B={1, \ldots,|\mathcal{P}|}$ and define $f: A \rightarrow B$ by letting $f(a)$ be the number of the block that contains $A$.

Where did the rather strange name “Pigeonhole Principle” come from? Old style desks often had what looked like a stacked array of boxes that were open in the front. These boxes were usually used to hold various letters and folded or rolled papers. The boxes were called pigeon holes because of their resemblance to nesting boxes in pigeon coops. If $|A|$ letters are placed in $|B|$ pigeonholes in a desk and $|A|>|B|$, then at least one pigeonhole contains at least two documents.

## 数学代写|组合学代写Combinatorics代考|Boolean Functions

A Boolean function $f$ is a map from ${0,1}^n$ to ${0,1}$. Thus the domain of $f$ is all $n$ long vectors of zeroes and ones. Boolean functions arise in logic, where 0 is often replaced by $F$ for “False” and 1 by $T$ for “True.” Boolean functions also arise in arithmetic, where 0 and 1 are digits of numbers in binary representation. Mathematically, there is no difference between these interpretations; however, the two different interpretations have slightly different notation associated with them.

Example 2.17 Basic Boolean functions Here are three functions from ${0,1}^2$ to ${0,1}$ in two-line form

If we think of $x$ and $y$ as integers, we can write $p(x, y)=x y$ and, indeed, this is the notation that is commonly used for $p$. To emphasize the multiplication, we might write $p(x, y)=x \cdot y$. Suppose that $X$ and $Y$ are statements; e.g., $X$ may be the statement “It is cloudy.” and $Y$ may be “It is hot.” We can build a more complicated statement from these two in many simple ways. One possibility is “It is cloudy and it is hot.”
We could abbreviate this to ” $X$ and $Y$.” Let this compound statement be $Z$. Let $x=0$ if $X$ is false and $x=1$ if $X$ is true. Define $y$ and $z$ similarly. (This is the True/False interpretation of 0 and 1 mentioned earlier.) You should be able to see that $z=p(x, y)$ because $Z$ is true if and only if both $X$ and $Y$ are true. Not surprisingly, the function $p$ is called and in logic. Logicians sometimes write $p(x, y)=x \wedge y$ instead of $p(x, y)=x y$

## 数学代写|组合学代写Combinatorics代考|The Pigeonhole Principle

“鸽子洞原理”这个奇怪的名字从何而来?老式的桌子通常有一堆盒子，这些盒子在前面是打开的。这些盒子通常用来装各种信件和折叠或卷起来的文件。这些盒子被称为鸽子洞，因为它们与鸽舍里的筑巢盒很像。如果将$|A|$字母分别放在书桌的$|B|$和$|A|>|B|$中，则至少有一个鸽子洞包含至少两个文档。

## 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。