Course: 16.1 声明和声明 | The Way To Learn

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和 Or 语句

The words “and” and “or” are common in everyday language. In mathematics, there are some subtle differences that you need to watch out for, especially considering the word “or”.
::“和”和“或”在日常语言中是常见的,在数学中,有一些微妙的差异需要你注意,特别是考虑到“或”一词。

It will rain or it will snow.
::它会下雨或下雪。

When is this statement true and when is it false?
::这一声明何时真实,何时虚假?

Introduction to Logic
::逻辑引言

An atomic statement is a declarative statement without logical connectives that has a truth value . A logical connective is a word or symbol that joins two atomic statements to form a larger logical statement. Here are two declarative statements that are atomic statements:
::原子语句是一种没有逻辑关联的、具有真理价值的宣示性声明。逻辑关联性是一个词或符号,它结合两个原子语句形成一个更大的逻辑语句。下面是两个宣告性声明,它们是原子语句:

$\begin{aligned} P = I t i s s n o w i n g \end{aligned}$ .
::下雪了下雪了下雪了下雪了下雪了下雪了下雪了下雪了下雪了下雪了下雪了下雪了

$\begin{aligned} Q = I a m c o l d \end{aligned}$ .
::我很冷,我有点冷 I am cold.

The truth value of a statement is whether the statement is true or false. As a mathematician, your job is to determine when a logical statement is true and when it is false. If you don’t have enough information to determine whether the original statements are true or false, you can build a truth table to organize all the possible cases.
::声明的真理价值是声明是真实的还是虚假的。作为数学家,你的任务是确定逻辑声明是真实的还是虚假的。如果你没有足够的信息来确定原始声明是真实的还是虚假的,你可以建立一个真相表来组织所有可能的案件。

Consider the atomic statement $\begin{aligned} P \end{aligned}$ joined with the atomic statement $\begin{aligned} Q \end{aligned}$ . The following sentence can be written using the symbol “ $\begin{aligned} \lor \end{aligned}$ ” for the logical connective “or”.
::考虑原子声明P与原子声明Q合并。以下一句可以用逻辑连接“或”的“”符号来写。

\begin{aligned} I t i s s n o w i n g & o r I a m c o l d . \\ P & \lor Q \end{aligned}

::下雪了还是我冷了

This statement is a little strange because it seems to imply that it is always the case that one or both of those atomic statements is happening. Your common sense may dictate that this statement isn’t true because of course there are times when it is sunny and you are warm. It’s important to remember that not all statements are true! Your job is to determine what has to be true for the above statement to be true. To organize your work, you should construct a truth table. A truth table considers all possible combinations of the original atomic statements being true or false, and then uses logic to deduce the truth value of the compound statement in each case.
::这种说法有点奇怪,因为它似乎暗示,总是发生其中一种或两种原子声明的情况。你的常识可能表明,这一声明是不真实的,因为当然有些时候是阳光明媚的,而且你很温暖。重要的是要记住,并非所有声明都是真实的!你的任务是确定上述声明的真相。为了组织你的工作,你应该构建一个真相表。真相表将原原子声明的所有可能的组合都视为真实的或虚假的,然后用逻辑来推断每起案件复合声明的真相价值。

Here is the truth table for OR:
::以下是OR的真相表:

$\begin{aligned} P \end{aligned}$	$\begin{aligned} Q \end{aligned}$	$\begin{aligned} P \lor Q \end{aligned}$
T	T	T
T	F	T
F	T	T
F	F	F

Notice that there are four possible truth combinations of $\begin{aligned} P \end{aligned}$ and $\begin{aligned} Q \end{aligned}$ (both true, first true/second false, first false/second true, both false). Only one of these combinations yields a false statement for $\begin{aligned} P \lor Q \end{aligned}$ . What this means is that the statement “It is snowing or I am cold” is only false if “it is snowing” is false and “I am cold” is false. Note that if “it is snowing” is true and “I am cold” is also true, then “It is snowing or I am cold” is true. In mathematics, the word “or” does not mean exactly one or the other. It means “one or the other or both”.
::请注意,P和Q之间可能有四种可能的真相组合(两种情况都是真实的,先是真实的,先是真实的/第二次的,后是虚假的,两者都是虚假的)。只有一种情况组合为P提出虚假的陈述。这意味着“下雪或我寒冷”的说法只有在“下雪”是虚假的,“我冷”是虚假的。请注意,如果“下雪”是真实的,而“我冷”也是真实的,那么“下雪或我冷”是真实的。在数学中,“或”一词不完全指一个或另一个,意思是“一个或另一个或两个”。

Next consider the truth table for the following statement that uses the connective “and”. The following sentence can be written using the symbol “ $\begin{aligned} \land \end{aligned}$ ” for the logical connective “and”.
::下面考虑使用连接“和”的下列语句的“和”的“和”的真相表。以下句子可以用逻辑连接“和”的“和”的“和”符号来写。

\begin{aligned} I t i s s n o w i n g & a n d I a m c o l d . \\ P & \land Q \end{aligned}

::下雪了,我又冷

Here is the truth table for AND:
::以下是真相表:

$\begin{aligned} P \end{aligned}$	$\begin{aligned} Q \end{aligned}$	$\begin{aligned} P \land Q \end{aligned}$
T	T	T
T	F	F
F	T	F
F	F	F

Notice that a compound statement using “and” is true only if each atomic statement is individually true.
::注意使用“和”的复式说明只有在每个原子说明都符合个别情况时才属属实。

Watch the portion of this video focusing on of conjunction and disjunction :
::观看这段影片的片段,

Examples
::实例

Example 1
::例1

Earlier, you were asked when the statement "It will rain or it will snow" will be true and false. In English, most people use the word “or” to mean exclusive “or”. If you were told “you can have a brownie or a cookie for dessert”, you would assume you had to choose just one and couldn’t have both the brownie and the cookie. In mathematics, the word “or” means “one or the other or both”. Therefore in logic, “or” includes the case when both atomic parts of the state are true.
::早些时候,有人问到“下雨或下雪”的说法是真实的和假的。在英语中,大多数人用“或”一词来表示独家的“或”的意思。如果你被告知“你可以有一块布朗尼或饼干做甜点 ” , 你就会认为你只得选择一个,不能同时有布朗尼和饼干。在数学中,“或”一词的意思是“一个或两个或两个 ” 。因此,逻辑上,“或”包括国家的两个原子部分都是真实的。

$\begin{aligned} P = I t w i l l r a i n \end{aligned}$ .
::雨会下雨的

$\begin{aligned} Q = I t w i l l s n o w \end{aligned}$ .
::会下雪的

$\begin{aligned} P \lor Q \end{aligned}$
::

$\begin{aligned} P \end{aligned}$	$\begin{aligned} Q \end{aligned}$	$\begin{aligned} P \lor Q \end{aligned}$
T	T	T
T	F	T
F	T	T
F	F	F

The statement is only false when both parts of the statement are false. In other words, the statement is only false if “it will rain” is false and “it will snow” is also false. When one or both parts of an “or” statement are true then the whole statement is true.
::声明只有在声明的两部分都是假的时才是假的,换句话说,声明只有在“雨会下雨”是假的而且“雪会下雪”也是虚假的情况下才是虚假的。当“或”声明的一个或两个部分都是真的时,整个声明都是真实的。

Example 2
::例2

Identify the atomic statements in the following compound sentence. Then, use logical connectives to rewrite the sentence with symbols.
::识别以下复句中的原子语句。然后, 使用逻辑连接来用符号重写句子。

I am tired and hungry and I want a burger or a nap.
::我累了又饿,我想吃汉堡或者睡午觉

The proper way to interpret this sentence is to identify the “or” as relating to just the burger and the nap.
::解释这一句的适当方式是确定“或”一词仅与汉堡和午睡有关。

$\begin{aligned} P = I a m t i r e d \end{aligned}$ .
::P=我累了。 \ NP=我累了。

$\begin{aligned} Q = I a m h u n g r y \end{aligned}$ .
::我饿了

$\begin{aligned} R = I w a n t a b u r g e r \end{aligned}$ .
::我想吃个汉堡

$\begin{aligned} S = I w a n t a n a p \end{aligned}$ .
::我想睡个觉

The sentence could be rewritten with symbols as: $\begin{aligned} (P \land Q) \land (R \lor S) \end{aligned}$
::句子可以用以下符号重写P)(RS)

Example 3
::例3

For lunch you had a ham and cheese sandwich and an apple or an orange.
::午餐时你吃了火腿和奶酪三明治还有苹果或橙子

Not all sentences will be easy to break down into atomic statements. In this case, the ham and cheese sandwich is inseparable even though it contains the word “and”. You have to use your prior knowledge to know that “ham and cheese sandwich” is a type of sandwich.
::并非所有的句子都很容易细分为原子语句。在这种情况下,火腿和奶酪三明治是不可分割的,尽管它含有“和”一词。你必须利用以前的知识知道“火腿和奶酪三明治”是一种三明治。

$\begin{aligned} A = Y o u h a d a h a m a n d c h e e s e s a n d w i c h f o r l u n c h \end{aligned}$ .
::你午餐吃火腿和奶酪三明治

$\begin{aligned} B = Y o u h a d a n a p p l e f o r l u n c h \end{aligned}$ .
::你午餐吃了个苹果

$\begin{aligned} C = Y o u h a d a n o r a n g e f o r l u n c h \end{aligned}$ .
::你午餐吃了橙子

The sentence could be rewritten with symbols as: $\begin{aligned} A \land (B \lor C) \end{aligned}$ .
::句子可以用A(BC)的符号重写。

Note that each statement $\begin{aligned} A, B, C \end{aligned}$ contains the words “you had” and “for lunch” and is a complete sentence.
::请注意,每一份说明A、B、C都载有“你”和“午餐”两字,是一句完整的句子。

Example 4
::例4

Diagram the sentence from Guided Practice #1 using the logical connectives “ $\begin{aligned} \lor \end{aligned}$ ” for “or” and “ $\begin{aligned} \land \end{aligned}$ ” for “and”.
::“或”和“和”的逻辑连接词“___”和“___”用逻辑连接词“___”和“和”。

The hardest part in diagramming the logical connectives is often determining which parts of the sentence should be grouped together. In this case there is a clear separation between the three positive outcomes and with the two negative outcomes:
::逻辑连接图解的最难部分往往是确定该句的哪些部分应归为一组。在这种情况下,三个正结果和两个负结果之间有明确的区分:

$\begin{aligned} (M \land B \land D) \lor (W \land J) \end{aligned}$
:MBD) (WJ)

Example 5
::例5

Use a truth table to identify all cases when the statement in Guided Practice #1 is true or false.
::使用真相表来查明所有符合或不符合第1号《实践指南》声明的情况。

Truth tables of complex sentences can be overwhelming, especially since 5 atomic statements means that there should be $\begin{aligned} 2^{5} \end{aligned}$ rows in the truth table to account for all of the T/F combinations. To save time and space you can note that the statement $\begin{aligned} M \land B \land D \end{aligned}$ is only true when $\begin{aligned} M, B, \end{aligned}$ and $\begin{aligned} D \end{aligned}$ are all true and $\begin{aligned} W \land J \end{aligned}$ is only true when both $\begin{aligned} W \end{aligned}$ and $\begin{aligned} J \end{aligned}$ are true. This means that you now only need 4 rows in the truth table.
::复杂句子的真相表格可能令人难以接受,特别是因为五个原子语句意味着真相表格中应该有25行来说明所有T/F组合。为了节省时间和空间,你可以注意到,只有当M、B和D都属实时,MBD的语句才是真实的,WJ只有在W和J都属实时才是真实的。这意味着你现在只需要在真相表格中写4行。

$\begin{aligned} M \land B \land D \end{aligned}$	$\begin{aligned} W \land J \end{aligned}$	$\begin{aligned} (M \land B \land D) \lor (W \land J) \end{aligned}$
T	T	T
F	T	T
T	F	T
F	F	F

The statement is true if:
::声明是真实的,如果:

$\begin{aligned} M, B, \end{aligned}$ and $\begin{aligned} D \end{aligned}$ are all true.
::M,B,和D都是真实的。

$\begin{aligned} W \end{aligned}$ and $\begin{aligned} J \end{aligned}$ are both true.
::W和J都是真实的

$\begin{aligned} M, B, D, W, \end{aligned}$ and $\begin{aligned} J \end{aligned}$ are all true.
::M,B,D,W,和J都是真实的。

The statement is false if:
::声明是虚假的,如果:

Not all of $\begin{aligned} M, B, \end{aligned}$ and $\begin{aligned} D \end{aligned}$ are true and not both of $\begin{aligned} W \end{aligned}$ and $\begin{aligned} J \end{aligned}$ are true.
::不是所有的M、B和D都是真实的, W和J也不都是真实的。

Summary

The truth value of a statement is whether the statement is true or false.
::声明的真相价值在于声明是真实的还是虚假的。

A logical connective is a word that joins two declarative statements together. Two logical connectives are AND “ $\begin{aligned} \land \end{aligned}$ ” and OR “ $\begin{aligned} \lor \end{aligned}$ ”
::逻辑联系是指将两个声明合并在一起的词。两种逻辑联系是“ ” 和“ ” 或“ ” 。

An atomic statement is a declarative statement without logical connectives that has a truth value. These statements are signified by capital letters, such as P or Q.
::原子声明是一种宣言性声明,没有具有真理价值的逻辑联系,这些声明用大写字母(如P或Q)表示。

Review
::回顾

I go to school and do my work or stay home and play games.
::我上学做我的工作或者呆在家里玩游戏

1. Identify the atomic statements in the above compound sentence.
::1. 标明上述复数句中的原子说明。

2. Use logical connectives to rewrite the sentence with symbols.
::2. 使用逻辑连接来用符号重写句子。

I have macaroni and cheese or steak and green beans or potatoes.
::我有通心粉奶酪牛排绿豆土豆

3. Identify the atomic statements in the above compound sentence.
::3. 标明上述复数句中的原子说明。

4. Use logical connectives to rewrite the sentence with symbols.
::4. 使用逻辑连接来用符号重写句子。

I wear flip flops and either shorts and a t-shirt or a dress.
::我穿翻滚裤穿短裤穿T恤穿裙子

5. Identify the atomic statements in the above compound sentence.
::5. 标明上述复数句中的原子说明。

6. Use logical connectives to rewrite the sentence with symbols.
::6. 使用逻辑连接来用符号重写句子。

It is dark outside and I light a candle.
::外面一片漆黑我点了根蜡烛

7. Identify the atomic statements in the above compound sentence.
::7. 标明上述复数句中的原子说明。

8. Use logical connectives to rewrite the sentence with symbols.
::8. 使用逻辑连接来用符号重写句子。

We will go to the beach and have a picnic or go to the movies and eat popcorn.
::我们去海滩野餐或者去看电影吃爆米花

9. Identify the atomic statements in the above compound sentence.
::9. 标明上述复数句中的原子说明。

10. Use logical connectives to rewrite the sentence with symbols.
::10. 使用逻辑连接来用符号重写句子。

Make a truth table for each of the following statements.
::为以下每一声明绘制一个真相表。

11. $\begin{aligned} (P \land Q) \lor R \end{aligned}$
::11. (P) R

12. $\begin{aligned} P \land (Q \lor R) \end{aligned}$
::12. P____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

13. $\begin{aligned} (P \lor Q) \lor R \end{aligned}$
::13. (P) R

14. $\begin{aligned} P \lor (Q \lor R) \end{aligned}$
::14. P____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

15. How does the placement of parentheses affect the truth values of compound statements?
::15. 括号的放置如何影响复式发言的真伪价值?

Review (Answers)
::回顾(答复)

Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键。

Section outline

General

和 Or 语句

Introduction to Logic ::逻辑引言

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Review ::回顾

Review (Answers) ::回顾(答复)