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# 数学代写|离散数学代写Discrete Mathematics代考|MA2201/CS2022 Logical Form and Logical Equivalence

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## 数学代写|离散数学代写Discrete Mathematics代考|Logical Form and Logical Equivalence

Logic is a science of the necessary laws of thought, without which no employment of the understanding and the reason takes place. -Immanuel Kant, 1785
An argument is a sequence of statements aimed at demonstrating the truth of an assertion. The assertion at the end of the sequence is called the conclusion, and the preceding statements are called premises. To have confidence in the conclusion that you draw from an argument, you must be sure that the premises are acceptable on their own merits or follow from other statements that are known to be true.

In logic, the form of an argument is distinguished from its content. Logical analysis won’t help you determine the intrinsic merit of an argument’s content, but it will help you analyze an argument’s form to determine whether the truth of the conclusion follows necessarily from the truth of the premises. For this reason logic is sometimes defined as the science of necessary inference or the science of reasoning.

Consider the following two arguments. They have very different content but their logical form is the same. To help make this clear, we use letters like $p, q$, and $r$ to represent component sentences; we let the expression “not $p$ ” refer to the sentence “It is not the case that $p$ “; and we let the symbol $\therefore$ stand for the word “therefore.”
Argument 1
If the bell rings or the flag drops, then the race is over.
$\therefore$ If the race is not over, then $\overbrace{\text { the bell hasn’t rung and the flag hasn’t dropped. }}^{\text {not } r}$ not $q$.

## 数学代写|离散数学代写Discrete Mathematics代考|Compound Statements

We now introduce three symbols that are used to build more complicated logical expressions out of simpler ones. The symbol denotes not, $\wedge$ denotes and, and $\vee$ denotes or. Given a statement $p$, the sentence ” $\sim p$ ” is read “not $p$ ” or “It is not the case that $p$ “. In some computer languages the symbol $\neg$ is used in place of $\sim$. Given another statement $q$, the sentence ” $p \wedge q$ ” is read ” $p$ and $q$.” The sentence ” $p \vee q$ ” is read ” $p$ or $q$.”

In expressions that include the symbol $\sim$ as well as $\wedge$ or $\vee$, the order of operations specifies that $\sim$ is performed first. For instance, $\sim p \wedge q=(\sim p) \wedge q$. In logical expressions, as in ordinary algebraic expressions, the order of operations can be overridden through the use of parentheses. Thus $\sim(p \wedge q)$ represents the negation of the conjunction of $p$ and $q$. In this, as in most treatments of logic, the symbols $\wedge$ and $\vee$ are considered coequal in order of operation, and an expression such as $p \wedge q \vee r$ is considered ambiguous. This expression must be written as either $(p \wedge q) \vee r$ or $p \wedge(q \vee r)$ to have meaning.

A variety of English words translate into logic as $\wedge, \vee$, or $\sim$. For instance, the word but translates the same as and when it links two independent clauses, as in “Jim is tall but he is not heavy.” Generally, the word but is used in place of and when the part of the sentence that follows is, in some way, unexpected. Another example involves the words neither-nor. When Shakespeare wrote, “Neither a borrower nor a lender be,” he meant, “Do not be a borrower and do not be a lender.” So if $p$ and $q$ are statements, then
\begin{tabular}{|rrl|}
\hline$p$ but $q$ & means & $p$ and $q$ \
neither $p$ nor $q$ & means & $\sim p$ and $\sim q$ \
\hline
\end{tabular}

## 数学代写|离散数学代写离散数学代考|逻辑形式与逻辑等价

$\therefore$如果比赛还没有结束，那么$\overbrace{\text { the bell hasn’t rung and the flag hasn’t dropped. }}^{\text {not } r}$不是$q$。

## 数学代写|离散数学代写离散数学代考|复合语句

.

\begin{tabular}{|rrl|}
\hline$p$ but $q$ & means & $p$ and $q$ \
neither $p$ nor $q$ & means & $\sim p$ and $\sim q$ \
\hline
\end{tabular}

## 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。