离散数学 学习笔记(一) : 基础概念

proposition(命题): A proposition is a declarative sentence (that is, a sentence that declares a fact) that is either true or false, but not both.
设计命题的逻辑领域称为: 命题演算 或 命题推理. 由亚里士多德创建的.
The truth value of a proposition is true, denoted by T, if it is a true proposition; the truth value of a proposition is false, denoted by F, if it is a false proposition.

• 定义1: Let p be a proposition. The negation of (p), denoted by$ eg{p} $ (also denoted by $ ar{p} $), is the statement
  “It is not the case that (p).”
  The proposition $ eg{p} $ is read “not p.” The truth value of the negation of $ p $, $ eg{p} $, is the opposite of the truth value of (p).
• 定义2: Let (p) and (q) be propositions. The conjunction(合取) of (p) and (q), denoted by ({p} wedge {q}), is the proposition “(p) and (q).” The conjunction ({p} wedge {q}) is true when both (p) and (q) are true and is false otherwise.
• 定义3: Let (p) and (q) be propositions. The disjunction(析取) of (p) and (q), denoted by ({p} vee {q}), is the proposition “(p) and (q).” The disjunction ({p} vee {q}) is false when both (p) and (q) are false and is true otherwise.
  
• 定义4: Let (p) and (q) be propositions. The exclusive or of (p) and (q), denoted by $ {p} igoplus {q}$, is the proposition that is true when exactly one of (p) and (q) is true and is false otherwise.
  inclusive or (同或) : 相同为T, 不同为F
  exclusive or (异或) : 相同为F, 不同为T
• 定义5: Let (p) and (q) be propositions. The conditional statement(条件陈述) (p o q) is the proposition “if (p), then (q).” The conditional statement (p) and (q) is false when (p) is true and (q) is false, and true otherwise.
  In the conditional statement (p o q), (p) is called the hypothesis(假设) (or antecedent or premise) and (q) is called the conclusion(结论) (or consequence).
  A conditional statement is also called an implication

CONVERSE(逆), CONTRAPOSITIVE(倒置), AND INVERSE(反)
逆蕴含: The proposition ({q} o {p}) is called the converse of ({p} o {q}).
倒置蕴含: The contrapositive of ({p} o {q}) is the proposition ( eg{q} o eg{p}).
反蕴含: The proposition ( eg{p} o eg{q}) is called the inverse of ({p} o {q}).

只有倒置蕴含与 ({p} o {q})由相同的真值.
无论 (p) 和 (q) 真值如何, 逆蕴含和反蕴含都与({p} o {q}) 没用相同的真值.
当两个复合命题总是具有相同真值, 称之为等价.一个蕴含与其倒置蕴含等价; 一个蕴含的逆蕴含与反蕴含也是等价的.