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# 数学代写|离散数学代写Discrete Mathematics代考|MATH215 Order of Quantifiers

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## 数学代写|离散数学代写Discrete Mathematics代考|Order of Quantifiers

Consider the following two statements:
$\forall$ person $x, \exists$ a person $y$ such that $x$ loves $y$. $\exists$ a person $y$ such that $\forall$ person $x, x$ loves $y$.
Note that except for the order of the quantifiers, these statements are identical. However, the first means that given any person, it is possible to find someone whom that person loves, whereas the second means that there is one amazing individual who is loved by all people. (Reread the statements carefully to verify these interpretations!) The two sentences illustrate an extremely important property about statements with two different quantifiers.
In a statement containing both $\forall$ and $\exists$, changing the order of the quantifiers can significantly change the meaning of the statement.

Interestingly, however, if one quantifier immediately follows another quantifier of the same type, then the order of the quantifiers does not affect the meaning. Consider the commutative property of addition of real numbers, for example:
$\forall$ real number $x$ and $\forall$ real number $y, x+y=y+x$.
This means the same as
$\forall$ real number $y$ and $\forall$ real number $x, x+y=y+x$.
Thus the property can be expressed a little less formally as
$\forall$ real numbers $x$ and $y, x+y=y+x$.

## 数学代写|离散数学代写Discrete Mathematics代考|Quantifier Order in a Tarski World

Look again at the Tarski world of Figure 3.3.1. Do the following two statements have the same truth value?
a. For every square $x$ there is a triangle $y$ such that $x$ and $y$ have different colors.
b. There exists a triangle $y$ such that for every square $x, x$ and $y$ have different colors.
Solution Statement (a) says that if someone gives you one of the squares from the Tarski world, you can find a triangle that has a different color. This is true. If someone gives you square $g$ or $h$ (which are gray), you can use triangle $d$ (which is black); if someone gives you square $e$ (which is black), you can use either triangle $f$ or $i$ (which are gray); and if someone gives you square $j$ (which is blue), you can use triangle $d$ (which is black) or triangle $f$ or $i$ (which are gray).

Statement (b) says that there is one particular triangle in the Tarski world that has a different color from every one of the squares in the world. This is false. Two of the triangles are gray, but they cannot be used to show the truth of the statement because the Tarski world contains gray squares. The only other triangle is black, but it cannot be used either because there is a black square in the Tarski world.

Thus one of the statements is true and the other is false, and so they have opposite truth values.

## 数学代写|离散数学代写|离散数学代考|量词的顺序

$forall$人$x, 存在$y$这样的人，$x$爱$y。\存在着一个人$y$，使$x$爱着$y$。
$y$.

$forall$实数$x$和$forall$实数$y，x+y=y+x$。

$forall$实数$y$和$forall$实数$x，x+y=y+x$。

$对于所有$实数$x$和$y，x+y=y+x$。

## 数学代写|离散数学代写Discrete Mathematics代考|Tarski世界中的量化器秩序

a. 对于每一个正方形$x$，都有一个三角形$y$，使得$x$和$y$有不同的颜色。
b. 存在一个三角形$y$，使得对于每个正方形$x，x$和$y$有不同的颜色。

## 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。