Logic and Proofs--离散数学

1. Propositions: A proposition is a declarative sentence(that is, a sentence that declares a fact ) that is either true or false, but not both.

命题是一个陈述句（即陈述事实的句子），它或真或假，但不能既真又假。

1. When Alexander the Great died in 323 B.C.E, a backlash against anything related to Alexander led to trumped-up charges of impiety against Aristotle. Aristotle fled to Chalcis to avoid prosecution. He only lived one year in Chalcis, dying of a stomach ailment in 322 B.C.E.

当亚历山大大帝于公元前323年去世后，那里立刻掀起了反亚历山大的狂潮，致使亚里士多德被冠以莫须有的不敬神罪名。亚里士多德逃亡到加而西斯避难。他在加而西斯生活了一年，于公元前322年死于胃病。

1. Let p be a proposition. The negation of p, denoted by ¬p, is the statement

“It is not the case that p.”

The proposition ¬p is read “not p.” The truth value of the negation of p, ¬p, is opposite of the truth value of p.

令p为一命题，则p的否定记作¬p，指

“不是p所指的情形。”

命题¬p读作“非p”。p的否定（¬p）的真值和p的真值相反。

1. Let p and q be propositions. The conjunction of p and q, denoted by p∧q, is the proposition “p and q.” The conjunction p∧q is true when both p and q are true and is false otherwise.

Note that in logic the word “but” sometimes is used instead of “and” in a conjunction. For example, the statement “The sun is shining, but it is raining” is another way of saying “The sun is shining and it is raining.” (In natural language, there is a subtle difference in meaning between “and” and “but”; we will not be concerned with this nuance here.)

令p和q为命题。p、q的合取即命题“p并且q”，记作p∧q。当p和q都是真时，p∧q命题为真，否则为假。

注意在逻辑合取中，有时候用到“但是”一词，而非“并且”一词。比如，语句“阳光灿烂，但是在下雨”是“阳光灿烂并且在下雨”一句的另一种说法。（在自然语言中，“并且”和“但是”在意思上有微妙的不同，这里我们不关心这个细微差别。）

1. Let p and p be propositions. The disjunction of p and q, denoted by p∨q, is the proposition “p or q.” The disjunction p∨q is false when both p and q are false and is true otherwise.

The use of the connective or in a disjunction corresponds to one of the two ways the word or is used in English, namely, in an inclusive way. Thus, a disjunction is true when at least one of the two propositions in it is true. Sometimes, we use or in an exclusive sense. When the exclusive or is used to connect the propositions p and q, the proposition “p or q (but not both)” is obtained. This proposition is true when p is true and q is false, and when p is false and q is true. It is false when both p and q are false and when both are true.

Let p and q be propositions. The exclusive or of p and q, denoted by p⊕q, is the proposition that is true when exactly one of p and q is true and is false otherwise.

令p和q为命题。P和q的析取式即命题“p或q”，记作p∨q。当p和q均为假时，析取命题p∨q为假，否则为真。

在析取中使用的联结词或对应于自然语言中或的两种情形之一，即可兼得的。这样，当析取中的两个命题之中至少有一个为真时，析取为真。有时我们也按不可兼得的方式使用或。当用异或来联结命题p和q时，就得到命题“p或q（但非两者）”。这一命题当p为真且q为假时为真，并且当p为假且q为真时也为真，而当p和q两者均为假或均为真时，这一命题为假。

1. Let p and q be propositions. The conditional statement p->q is the proposition “if p, then q.” The conditional statement p->q is false when p is true and q is false, and true otherwise. In the conditional statement p->q, p is called the hypothesis (or antecedent or premise) and q is called the conclusion (or consequence).

在条件语句p->q中，p称为假设（前件、前提），q称为结论（后件）。

The statement p->q is called a conditional statement because p->q asserts that q is true on the conditional that p holds. A conditional statement is also called an implication.

语句p->q称为条件语句，因为p->q可以断定在条件p成立的时候q为真。条件语句也称为蕴含。

Because conditional statements play such an essential role in mathematical reasoning, a variety of terminology is used to express p->q. You will encounter most if not all of the following ways to express this conditional statement:

由于条件语句在数学推理中具有很重要的作用，所有表达p->q术语也很多。即使不是全部，你也会碰到下面几个常用的条件语句的表述方式：

“if p, then q”  如果p，则q         “p implies q”  p蕴含q

“if p, q”      如果p，q            “p only if q”  q仅当p

“p is sufficient for q” p是q的充分条件“a sufficient condition for q is p”  q的充分条件是q

“q if p”       q如果p             “q whenever p”  q每当p

“q when p”   q当p               “q is necessary for p”  q是p的必要条件

“a necessary condition for p is q” p的必要条件是q  “q follows from p”  q由p得出

“q unless ¬p ”  q除非¬p

Of the various ways to express the conditional statement p->q, the two that seem to cause the most confusion are “p only if q” and “q unless ¬p.” Consequently, we will provide some guidance for clearing up this confusion.

To remember that “p only if q” expresses the same thing as “if p, then q,” note that “p only if q” says that p cannot be true when q is not true. That is, the statement is false if p is true, but q is false. When p is false, q may be either true or false, because the statement says nothing about the truth value of q. Be careful not to use “q only if p” to express p->q because this is incorrect. To see this, note that the true values of “q only if p” and p->q are different when p and q have different truth values.

To remember that “q unless ¬p” expresses the same conditional statement as “if p, then q,” note that “q unless ¬p” means that if ¬p if false, then q must be true.

1. Let p and q be propositions. The biconditional statement p <-> q is the proposition “p if and only if q.” The biconditional statement p <-> p is true when p and q have the same truth values, and is false otherwise. Biconditional statements are also called bi-implications.

Biconditionals :双条件语句

Bi-implications :双向蕴含

There are some other common ways to express p <-> q :

“p is necessary and sufficient for q”  “p是q的充分必要条件”

“if p then q, and conversely”        “如果p那么q，反之亦然”

“p iff q”                         “p当且仅当q”

1. The proposition q->p is called the converse of p->q. The contrapositive of p->q is the proposition ¬q->¬p. The proposition ¬p->¬q is called the inverse of p->q.

A conditional statement and its contrapositive are equivalent. The converse and the inverse of a conditional statement are also equivalent, but neither is equivalent to the original conditional statement.

converse :逆命题

contrapositive :逆否命题

inverse :反命题

equivalent :等价

notation :记号

incorporate

quantified