介词at

介词或陈述 (Preposition or Statement)

A preposition is a definition sentence which is true or false but not both.

介词是一个定义语句，它是对还是错，但不能同时包含两者。

For example: The following 8 sentences,

例如：以下8个句子，

1. Paris in France

   法国巴黎

2. 2 + 2 =4

   2 + 2 = 4

3. London in Denmark

   丹麦伦敦

4. X = 2 is solution of x^2 = 4

   X = 2是x ^ 2 = 4的解

5. 1 + 1 = 2

   1 +1 = 2

6. 9<6

   9 <6

7. Where are you going?

   你要去哪里？

8. Do your homework

   做你的作业

All of them are preposition except vii and viii moreover i, ii and vi are true whereas iii, iv, v are false.

除了vii和viii之外，所有这些都是介词，而且i ， ii和vi是正确的，而iii ， iv和v是错误的。

复合命题 (Compound proposition)

Many propositions are composite that is composed of subpropositions and various connectives discussed subsequently. Such a composite proposition is said to be compound propositions. A proposition is called primitive if it cannot be broken down into the simpler proposition that is if it is not composite.

许多命题是复合的，由子命题和随后讨论的各种连接词组成。 这种复合命题被称为复合命题。 如果一个命题不能分解为更简单的命题，即不是复合命题，则该命题称为原始命题。

Example:

例：

1. "John intelligent or studies every night" is a compound proposition with subproposition. "John is intelligent" and "john studies every night".

   “约翰知识分子或每晚学习”是一个带有子命题的复合命题。 “约翰很聪明”和“约翰每晚学习” 。

2. "Roses are red and violets are blue" is a compound proposition with subproposition "Roses are red" and "violets are blue".

   “玫瑰是红色，紫罗兰是蓝色”是一个复合命题，子命题是“玫瑰是红色”和“紫罗兰是蓝色” 。

基本逻辑运算 (Basic logical operation)

The Three basic logical operations conjunction, disjunction, and negation which corresponds respectively. To the English words "and", "or" and "not".

的三个基本逻辑操作的同时 ， 析取 ，并否定其分别对应。 英文单词“ and” ， “ or”和“ not” 。

1) Conjunction (p ^ q):

1)连词(p ^ q)：

Any two proposition can be combined by the word and to form a compound proposition said to be the conjunction of the original proposition. Symbolically p ^ q read p and q denotes the conjunction of p and q. Since, p ^ q is a proposition it has the truth value and this truth value depends only on the truth values of p and q, specifically:

任何两个命题都可以用单词组合起来，形成一个复合命题，据说是原始命题的连词。 象征性p ^ Q读取p和q表示p和q的结合。 由于p ^ q是一个命题，因此它具有真值，并且该真值仅取决于p和q的真值，具体而言：

Definition: If p and q are true then p ^ q is true otherwise p ^ q is false.

定义：如果p和q为true，则p ^ q为true，否则p ^ q为false。

p	q	p ^ q
T	T	T
T	F	F
F	T	F
F	F	F

p	q	p ^ Q
Ť	Ť	Ť
Ť	F	F
F	Ť	F
F	F	F

Example: Consider the following 4 statements:

示例：考虑以下4条语句：

1. Paris is in France and 2+2 = 4

   巴黎在法国， 而 2 + 2 = 4

2. Paris is in France and 2 + 2 = 5

   巴黎是在法国和 2 + 2 = 5

3. Paris is in England and 2 + 2 = 4

   巴黎在英格兰， 而 2 + 2 = 4

4. Paris is in England and 2 + 2 = 5

   巴黎在英格兰， 而 2 + 2 = 5

In the given four statements only the first statement is true. Each of the other statements is false since at least one of its substatements is false.

在给定的四个语句中，只有第一个语句为true。 其他每个陈述都是假的，因为其至少一个子陈述是假的。

2) Disjunction (p V q)

2)取和(p V q)

Any two proposition can be combined by the word "or" to form a compound proposition is said to be the disjunction of the original proposition, symbolically p V q.

任何两个命题都可以由单词“或”组合成一个复合命题，据说这是原始命题的析取，符号为p V q 。

Read "p or q" denotes the disjunction of p and q. The truth value of p V q depends only on the truth values of p and q as follow:

读为“ p或q”表示p和q的析取。 p V q的真值仅取决于p和q的真值，如下所示：

Definition: If p and q are false then p V q is false, otherwise p V q is true.

定义：如果p和q为假，则p V q为假，否则p V q为真。

p	q	pVq
T	T	T
T	F	T
F	T	T
F	F	F

p	q	pVq
Ť	Ť	Ť
Ť	F	Ť
F	Ť	Ť
F	F	F

Example: Consider the following four statements:

示例：考虑以下四个语句：

1. Paris is in France or 2 + 2 = 4

   巴黎在法国或 2 + 2 = 4

2. Paris is in France or 2 + 2 = 5

   巴黎在法国或 2 + 2 = 5

3. Paris is in England or 2 + 2= 4

   巴黎在英格兰或 2 + 2 = 4

4. Paris is in England or 2 + 2 = 5

   巴黎在英格兰或 2 + 2 = 5

Only the last statements are false. Each of the other statements is true since at least of its substatements is true.

只有最后一个语句为假。 其他每个陈述都是正确的，因为至少其子陈述是正确的。

3) Negation( ~p)

3)否定(〜p)

Given any proposition p another proposition is said to be the negation of p can be formed by writing - it is not the case that... or "it is false that ...", before p or if possible by inserting in p the word "not" symbolically. ~p or ~p.

给定任何命题p，另一个命题可以说是p的否定可以通过书写来形成-在p之前或者如果可能的话，可以在p之前插入...或“ ... 是错误的...”的情况并非如此。单词“ not”象征性地。 〜p或〜p 。

Read "not p", denotes the negation of p. The truth value of p depends on the truth value of p as follows:

读为“ not p” ，表示p的否定 。 P的真值取决于p的真值，如下所示：

Definition: If p is true then ~p is false and if p is false then ~p is true.

定义：如果p为true，则〜p为false，如果p为false，则〜p为true。

p	~p
T	F
F	~F

p	〜p
Ť	F
F	〜F

翻译自: https://www.includehelp.com/basics/preposition-logic-in-discrete-mathematics.aspx

介词at