介詞at

介詞或陳述 (Preposition or Statement)

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

介詞是一個定義語句，它是對還是錯，但不能同時包含兩者。

For example: The following 8 sentences,

例如：以下8個句子，

1. Paris in France

   法國巴黎

2. 2 + 2 =4

   2 + 2 = 4

3. London in Denmark

   丹麥倫敦

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

   做你的作業

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，所有這些都是介詞，而且i ， ii和vi是正確的，而iii ， iv和v是錯誤的。

復合命題 (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.

許多命題是復合的，由子命題和隨后討論的各種連接詞組成。 這種復合命題被稱為復合命題。 如果一個命題不能分解為更簡單的命題，即不是復合命題，則該命題稱為原始命題。

Example:

例：

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

   “約翰知識分子或每晚學習”是一個帶有子命題的復合命題。 “約翰很聰明”和“約翰每晚學習” 。

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

   “玫瑰是紅色，紫羅蘭是藍色”是一個復合命題，子命題是“玫瑰是紅色”和“紫羅蘭是藍色” 。

基本邏輯運算 (Basic logical operation)

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

的三個基本邏輯操作的同時 ， 析取 ，并否定其分別對應。 英文單詞“ and” ， “ or”和“ not” 。

1) Conjunction (p ^ q):

1)連詞(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:

任何兩個命題都可以用單詞組合起來，形成一個復合命題，據說是原始命題的連詞。 象征性p ^ Q讀取p和q表示p和q的結合。 由于p ^ q是一個命題，因此它具有真值，并且該真值僅取決于p和q的真值，具體而言：

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

定義：如果p和q為true，則p ^ q為true，否則p ^ q為false。

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

p	q	p ^ Q
?	?	?
?	F	F
F	?	F
F	F	F

Example: Consider the following 4 statements:

示例：考慮以下4條語句：

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

   巴黎在法國， 而 2 + 2 = 4

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

   巴黎是在法國和 2 + 2 = 5

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

   巴黎在英格蘭， 而 2 + 2 = 4

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

   巴黎在英格蘭， 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.

在給定的四個語句中，只有第一個語句為true。 其他每個陳述都是假的，因為其至少一個子陳述是假的。

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.

任何兩個命題都可以由單詞“或”組合成一個復合命題，據說這是原始命題的析取，符號為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:

讀為“ p或q”表示p和q的析取。 p V q的真值僅取決于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.

定義：如果p和q為假，則p V q為假，否則p V q為真。

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

p	q	pVq
?	?	?
?	F	?
F	?	?
F	F	F

Example: Consider the following four statements:

示例：考慮以下四個語句：

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

   巴黎在法國或 2 + 2 = 4

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

   巴黎在法國或 2 + 2 = 5

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

   巴黎在英格蘭或 2 + 2 = 4

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

   巴黎在英格蘭或 2 + 2 = 5

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

只有最后一個語句為假。 其他每個陳述都是正確的，因為至少其子陳述是正確的。

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.

給定任何命題p，另一個命題可以說是p的否定可以通過書寫來形成-在p之前或者如果可能的話，可以在p之前插入...或“ ... 是錯誤的...”的情況并非如此。單詞“ 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:

讀為“ not p” ，表示p的否定 。 P的真值取決于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.

定義：如果p為true，則?p為false，如果p為false，則?p為true。

p	~p
T	F
F	~F

p	?p
?	F
F	?F

翻譯自: https://www.includehelp.com/basics/preposition-logic-in-discrete-mathematics.aspx

介詞at