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# 数学代写|离散数学代写Discrete Mathematics代考|‘‘Not”

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## 数学代写|离散数学代写Discrete Mathematics代考|Recalling the Strategy Definition

The statement “not $\mathbf{A}$,” written $\sim \mathbf{A}$, is true whenever $\mathbf{A}$ is false. For example, the statement
Charles is not happily married
is true provided the statement “Charles is happily married” is false. The truth table for $\sim \mathbf{A}$ is as follows:
\begin{tabular}{cc}
\hline $\mathbf{A}$ & $\sim \mathbf{A}$ \
\hline $\mathrm{T}$ & $\mathrm{F}$ \
$\mathrm{F}$ & $\mathrm{T}$ \
\hline
\end{tabular}
Greater understanding is obtained by combining the connectives:
EXAMPLE 1.6
We examine the truth table for $\sim(\mathbf{A} \wedge \mathbf{B})$ :
\begin{tabular}{lccc}
\hline $\mathbf{A}$ & $\mathbf{B}$ & $\mathbf{A} \wedge \mathbf{B}$ & $\sim(\mathbf{A} \wedge \mathbf{B})$ \
\hline $\mathrm{T}$ & $\mathrm{T}$ & $\mathrm{T}$ & $\mathrm{F}$ \
$\mathrm{T}$ & $\mathrm{F}$ & $\mathrm{F}$ & $\mathrm{T}$ \
$\mathrm{F}$ & $\mathrm{T}$ & $\mathrm{F}$ & $\mathrm{T}$ \
$\mathrm{F}$ & $\mathrm{F}$ & $\mathrm{F}$ & $\mathrm{T}$ \
\hline
\end{tabular}
EXAMPLE 1.7
Now we look at the truth table for $(\sim \mathbf{A}) \vee(\sim \mathbf{B})$ :
\begin{tabular}{ccccc}
\hline $\mathbf{A}$ & $\mathbf{B}$ & $\sim \mathbf{A}$ & $\sim \mathbf{B}$ & $(\sim \mathbf{A}) \vee(\sim \mathbf{B})$ \
\hline $\mathrm{T}$ & $\mathrm{T}$ & $\mathrm{F}$ & $\mathrm{F}$ & $\mathrm{F}$ \
$\mathrm{T}$ & $\mathrm{F}$ & $\mathrm{F}$ & $\mathrm{T}$ & $\mathrm{T}$ \
$\mathrm{F}$ & $\mathrm{T}$ & $\mathrm{T}$ & $\mathrm{F}$ & $\mathrm{T}$ \
$\mathrm{F}$ & $\mathrm{F}$ & $\mathrm{T}$ & $\mathrm{T}$ & $\mathrm{T}$ \
\hline
\end{tabular}

Notice that the statements $\sim(\mathbf{A} \wedge \mathbf{B})$ and $(\sim \mathbf{A}) \vee(\sim \mathbf{B})$ have the same truth table. As previously noted, such pairs of statements are called logically equivalent.
The logical equivalence of $\sim(\mathbf{A} \wedge \mathbf{B})$ with $(\sim \mathbf{A}) \vee(\sim \mathbf{B})$ makes good intuitive sense: the statement $\mathbf{A} \wedge \mathbf{B}$ fails [that is, $\sim(\mathbf{A} \wedge \mathbf{B})$ is true] precisely when either $\mathbf{A}$ is false or $\mathbf{B}$ is false. That is, $(\sim \mathbf{A}) \vee(\sim \mathbf{B})$. Since in mathematics we cannot rely on our intuition to establish facts, it is important to have the truth table technique for establishing logical equivalence. The exercise set will give you further practice with this notion.

One of the main reasons that we use the inclusive definition of “or” rather than the exclusive one is so that the connectives “and” and “or” have the nice relationship just discussed. It is also the case that $\sim(\mathbf{A} \vee \mathbf{B})$ and $(\sim \mathbf{A}) \wedge(\sim \mathbf{B})$ are logically equivalent. These logical equivalences are sometimes referred to as de Morgan’s laws.

## 数学代写|离散数学代写Discrete Mathematics代考|‘‘If-Then’’

A statement of the form “If $\mathbf{A}$ then $\mathbf{B}$ ” asserts that whenever $\mathbf{A}$ is true then $\mathbf{B}$ is also true. This assertion (or “promise”) is tested when $\mathbf{A}$ is true, because it is then claimed that something else (namely $\mathbf{B}$ ) is true as well. However, when $\mathbf{A}$ is false then the statement “If $\mathbf{A}$ then $\mathbf{B}$ ” claims nothing. Using the symbols $\mathbf{A} \Rightarrow \mathbf{B}$ to denote “If $\mathbf{A}$ then $\mathbf{B}$ “, we obtain the following truth table:
\begin{tabular}{ccc}
\hline $\mathbf{A}$ & $\mathbf{B}$ & $\mathbf{A} \Rightarrow \mathbf{B}$ \
\hline $\mathrm{T}$ & $\mathrm{T}$ & $\mathrm{T}$ \
$\mathrm{T}$ & $\mathrm{F}$ & $\mathrm{F}$ \
$\mathrm{F}$ & $\mathrm{T}$ & $\mathrm{T}$ \
$\mathrm{F}$ & $\mathrm{F}$ & $\mathrm{T}$ \
\hline
\end{tabular}
Notice that we use here an important principle of aristotelian logic: every sensible statement is either true or false. There is no “in between” status. When $\mathbf{A}$ is false we can hardly assert that $\mathbf{A} \Rightarrow \mathbf{B}$ is false. For $\mathbf{A} \Rightarrow \mathbf{B}$ asserts that “whenever A is true then $\mathbf{B}$ is true”, and $\mathbf{A}$ is not true!

Put in other words, when $\mathbf{A}$ is false then the statement $\mathbf{A} \Rightarrow \mathbf{B}$ is not tested. It therefore cannot be false. So it must be true. We refer to $\mathbf{A}$ as the hypothesis of the implication and to $\mathbf{B}$ as the conclusion of the implication. When the if-then statement is true, then the hypothsis implies the conclusion.
EXAMPLE 1.8
The statement “If $2=4$ then Calvin Coolidge was our greatest president” is true. This is the case no matter what you think of Calvin Coolidge. The point is that the hypothesis $(2=4)$ is false; thus it doesn’t matter what the truth value of the conclusion is. According to the truth table for implication, the sentence is true.
The statement “If fish have hair then chickens have lips” is true. Again, the hypothesis is false so the sentence is true.

The statement “If $9>5$ then dogs don’t fly” is true. In this case the hypothesis is certainly true and so is the conclusion. Therefore the sentence is true.
(Notice that the “if” part of the sentence and the “then” part of the sentence need not be related in any intuitive sense. The truth or falsity of an “if-then” statement is simply a fact about the logical values of its hypothesis and of its conclusion.)

## 数学代写|离散数学代写Discrete Mathematics代考|Recalling the Strategy Definition

\begin{tabular}{cc}
\hline $\mathbf{A}$ & $\sim \mathbf{A}$ \hline $\mathrm{T}$ & $\mathrm{F}$ \$\mathrm{F}$ & $\mathrm{T}$ \hline
\end{tabular}

\begin{tabular}{lccc}
\hline $\mathbf{A}$ & $\mathbf{B}$ & $\mathbf{A} \wedge \mathbf{B}$ & $\sim(\mathbf{A} \wedge \mathbf{B})$ \hline $\mathrm{T}$ & $\mathrm{T}$ & $\mathrm{T}$ & $\mathrm{F}$ \$\mathrm{T}$ & $\mathrm{F}$ & $\mathrm{F}$ & $\mathrm{T}$ \$\mathrm{F}$ & $\mathrm{T}$ & $\mathrm{F}$ & $\mathrm{T}$ \$\mathrm{F}$ & $\mathrm{F}$ & $\mathrm{F}$ & $\mathrm{T}$ \hline
\end{tabular}

\begin{tabular}{ccccc}
\hline $\mathbf{A}$ & $\mathbf{B}$ & $\sim \mathbf{A}$ & $\sim \mathbf{B}$ & $(\sim \mathbf{A}) \vee(\sim \mathbf{B})$ \hline $\mathrm{T}$ & $\mathrm{T}$ & $\mathrm{F}$ & $\mathrm{F}$ & $\mathrm{F}$ \$\mathrm{T}$ & $\mathrm{F}$ & $\mathrm{F}$ & $\mathrm{T}$ & $\mathrm{T}$ \$\mathrm{F}$ & $\mathrm{T}$ & $\mathrm{T}$ & $\mathrm{F}$ & $\mathrm{T}$ \$\mathrm{F}$ & $\mathrm{F}$ & $\mathrm{T}$ & $\mathrm{T}$ & $\mathrm{T}$ \hline
\end{tabular}

$\sim(\mathbf{A} \wedge \mathbf{B})$与$(\sim \mathbf{A}) \vee(\sim \mathbf{B})$的逻辑等价具有很好的直观意义:当$\mathbf{A}$为假或$\mathbf{B}$为假时，语句$\mathbf{A} \wedge \mathbf{B}$失败[即$\sim(\mathbf{A} \wedge \mathbf{B})$为真]。也就是$(\sim \mathbf{A}) \vee(\sim \mathbf{B})$。因为在数学中我们不能依靠我们的直觉来建立事实，所以有真值表技术来建立逻辑等价是很重要的。这个练习集将给你进一步练习这个概念。

## 数学代写|离散数学代写Discrete Mathematics代考|‘‘If-Then’’

“如果$\mathbf{A}$那么$\mathbf{B}$”这样的语句断言只要$\mathbf{A}$为真，那么$\mathbf{B}$也为真。当$\mathbf{A}$为真时，这个断言(或“承诺”)就会被检验，因为它随后就会声称其他东西(即$\mathbf{B}$)也为真。然而，当$\mathbf{A}$为假时，语句“如果$\mathbf{A}$那么$\mathbf{B}$”没有声明任何内容。用符号$\mathbf{A} \Rightarrow \mathbf{B}$表示“如果$\mathbf{A}$那么$\mathbf{B}$”，我们得到以下真值表:
\begin{tabular}{ccc}
\hline $\mathbf{A}$ & $\mathbf{B}$ & $\mathbf{A} \Rightarrow \mathbf{B}$ \hline $\mathrm{T}$ & $\mathrm{T}$ & $\mathrm{T}$ \$\mathrm{T}$ & $\mathrm{F}$ & $\mathrm{F}$ \$\mathrm{F}$ & $\mathrm{T}$ & $\mathrm{T}$ \$\mathrm{F}$ & $\mathrm{F}$ & $\mathrm{T}$ \hline
\end{tabular}

“如果$2=4$那么卡尔文·柯立芝就是我们最伟大的总统”这句话是对的。无论你怎么看待卡尔文·柯立芝，情况都是如此。关键是假设$(2=4)$是假的;因此结论的真值是多少并不重要。根据蕴涵真值表，句子为真。
“如果鱼有毛，那么鸡就有嘴唇”这句话是对的。假设为假，所以句子为真。

“如果$9>5$那么狗不会飞”这句话是对的。在这种情况下，假设当然是正确的，结论也是正确的。因此这个句子是真的。
(请注意，句子的“if”部分和句子的“then”部分不需要在任何直观意义上联系起来。一个”如果-那么”命题的真假仅仅是关于它的假设和结论的逻辑值的事实。

## MATLAB代写

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