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# 数学代写|离散数学代写Discrete Mathematics代考|CSC226 Direct Proof and Counterexample VI:Floor and Ceiling

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## 数学代写|离散数学代写Discrete Mathematics代考|Direct Proof and Counterexample VI:Floor and Ceiling

Proof serves many purposes simultaneously. In being exposed to the scrutiny and judgment of a new audience, [a] proof is subject to a constant process of criticism and revalidation. Errors, ambiguities, and misunderstandings are cleared up by constant exposure. Proof is respectability. Proof is the seal of authority.
Proof, in its best instances, increases understanding by revealing the heart of the matter. Proof suggests new mathematics. The novice who studies proofs gets closer to the creation of new mathematics. Proof is mathematical power, the electric voltage of the subject which vitalizes the static assertions of the theorems.
Finally, proof is ritual, and a celebration of the power of pure reason.
-Philip J. Davis and Reuben Hersh, The Mathematical Experience, 1981
Imagine a real number sitting on a number line. The floor and ceiling of the number are the integers to the immediate left and to the immediate right of the number (unless the number is an integer, in which case its floor and ceiling both equal the number itself). Many computer languages have built-in functions that compute floor and ceiling automatically. These functions are very convenient to use when writing certain kinds of computer programs. In addition, the concepts of floor and ceiling are important in analyzing the efficiency of many computer algorithms.
Definition
Given any real number $x$, the floor of $\boldsymbol{x}$, denoted $\lfloor x\rfloor$, is defined as follows:
$$\lfloor x\rfloor=\text { that unique integer } n \text { such that } n \leq x<n+1 .$$
Symbolically, if $x$ is a real number and $n$ is an integer, then
$$\lfloor x\rfloor=n \quad \Leftrightarrow \quad n \leq x<n+1 .$$

## 数学代写|离散数学代写Discrete Mathematics代考|Computing Floors and Ceilings

Compute $\lfloor x\rfloor$ and $\lceil x\rceil$ for each of the following values of $x$ :
a. $25 / 4$
b. $0.999$
c. $-2.01$
Solution
a. $25 / 4=6.25$ and $6<6.25<7$; hence $\lfloor 25 / 4\rfloor=6$ and $\lceil 25 / 4\rceil=7$.
b. $0<0.999<1$; hence $\lfloor 0.999\rfloor=0$ and $\lceil 0.999\rceil=1$.
c. $-3<-2.01<-2$; hence $\lfloor-2.01\rfloor=-3$ and $\lceil-2.01\rceil=-2$.

An Application
The 1,370 students at a college are given the opportunity to take buses to an out-of-town event. Each bus holds a maximum of 40 passengers.
a. For reasons of economy, the leader of the event will send only full buses. What is the maximum number of buses the event leader will send?
b. If the event leader is willing to send one partially filled bus, how many buses will be needed to allow all the students to take the trip?
Solution
a. $\lfloor 1370 / 40\rfloor=\lfloor 34.25\rfloor=34$
b. $\lceil 1370 / 40\rceil=\lceil 34.25\rceil=35$

## 数学代写|离散数学代写Discrete Mathematics代考|Direct Proof and Counterexample VI:Floor and Ceiling

-Philip J. Davis 和 Reuben Hersh，数学经验， 1981

$\lfloor x\rfloor=$ that unique integer $n$ such that $n \leq x<n+1$.

$$\lfloor x\rfloor=n \quad \Leftrightarrow \quad n \leq x<n+1 .$$

## 数学代写离散数学代写Discrete Mathematics代考|Computing Floors and Ceilings

b. $0.999$
c. $-2.01$

b. $0<0.999<1 ;$ 因此 $\lfloor 0.999\rfloor=0$ 和 $[0.999\rceil=1$.
C。 $-3<-2.01<-2$; 因此 $\lfloor-2.01\rfloor=-3$ 和 $\lceil-2.01\rceil=-2$.

$\mathrm{A}$ 。 出于经済原因，活动负责人将只派满载客车。活动负责人最多派多少辆巴士?
b. 如果活动负责人愿意派一辆部分满载的巴士，那么需要多少辆巴士才能让所有学生乘坐? 解决 方㝔
$\lfloor 1370 / 40\rfloor=\lfloor 34.25\rfloor=34$
b. $\lceil 1370 / 40\rceil=\lceil 34.25\rceil=35$

## 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。