2015离散数学命题及联结词解析

• Do you speak English?
--This is a question, not a statement.
• 这个人太有才了！ • Let’s go! • 我们要努力学习。 • Teenagers shouldn’t indulge in game.
-- It is not a statement, but an advice.
• A proposition can be represented by a proposition variable • A propositional variable is often denoted as p, q, r, etc. • e.g.
– p: Today is Tuesday. – q: 2+2=4 – r: 2+3=6
2018/10/11 25
How to make a truth table?
To create a truth table, follow these steps:
1. Determine the number of variables; for n variables, create a table with 2n rows.
2018/10/11 7
Logical Connectives
• A simple statement can be represented by an atom proposition. • More than one atom propositions can be combined into a compound statement. • The combination is achieved using “connectives”. Usually, the connective roughly corresponds some conjunctive in the natural language.
19
2018/10/11
The most confusion:
• “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 truth value of q. • example:
2018/10/11
2
Proposition
• Proposition (statement)
A declarative sentence (that is, a sentence that declares a fact) that is either true or false, but not both.
p T T q T F p q T F
(p q) ( q p)
F
F
T
F
F
T
2018/10/11
22
Biconditional
• Sentence:
– You can take the flight if and only if you buy a ticket. – 两个三角形全等，当且仅当它们的三组对 边相等。 • 表示的逻辑关系：p与q互为充分必要条件 “p is necessary and sufficient for q”
除非你努力，否则你将失败。
我今天进城，除非下雨。
2018/10/nd only if q” is denoted as p q, The statement is true
when p and have the same truth values , and is false otherwise . It is also called bi-implications.
• Truth value:
– The value (result) of the proposition(真值，是 命题最重要的属性） – One of the element from set {True, False}
2018/10/11 3
More Examples of Proposition
2018/10/11
10
Conjunction
The conjunction of p and q, denoted by pq ,is the proposition “p and q”
p T T q T F pq T F pq=true iff both p and q
F
F
T
F
F
F
All possible value of <p,q>
Propositional Logic
2018/10/11
1
Three tasks
1. What is a proposition?
2. Translate natural languages into logical expression. 3. How to make a truth table?
2018/10/11 11
合取联结词key word: 一边 …… 一边 …… 一面 …… 一面 …… 虽然…… 但是 …… 不但 ……而且 …… 既 ……又 …… 和 与 表示的逻辑关系是：p与q同时成立
2018/10/11 12
例:
(1) 2与3的最小公倍数是6。
(2)王丽和王娟是亲姐妹。
(3) 8能2整除，但不能被6整除。 (4) 2015年的春节下雪了 ,《我是歌手》很好看。
2018/10/11
18
Express conditional statement p q :
• • • • • • • • “if p,then q” • “p implies q” “if p,q” • “p only if q” “p is sufficient for q” • “q whenever p” “q if p” • “q follows from p” “q when p” “q unless p ” “a necessary condition for p is q” “a sufficient condition for q is p”
F
F
T
F
T
T
All possible value of <p,q>
2018/10/11 16
Conditional
• Sentence:
– If I am elected, then I will lower taxes. – If you get 100% on the final, then you will get an A. • 表示的逻辑关系是：q是p的必要条件
“if p then q, and conversely”
“p iff q”
2018/10/11 23
How to make a truth table?
What is wff(well-formed formula)?
定义1-3.1 命题演算的合式公式（wff） (1)单个命题变元本身是一个合式公式。 (2)如果A是合式公式，那么A是合式公式。 (3)如果A和B是合式公式，那么(AB), (AB), (AB), (A B)是合式公式。 (4)当且仅当能够有限次地应用(1)、(2)、(3)所得到 的包含命题变元、联结词和括号的符号串是合式公 式。
F
F
T
F
T
F
All possible value of <p,q>
2018/10/11 14
析取联结词key word: 或 表示的逻辑关系是：至少有一个成立
例：小王学过日语或俄语。
2018/10/11
15
Conditional
“if p, then q” is denoted as p q, asserts that q is true on the condition that p holds. p is called the hypothesis(or antecedent or premise ) and q is called the conclusion(or consequence). It is also called an implication. p q p q T T T F T F p q=false iff p is true and q is false
2018/10/11
5
More Examples of Proposition
有两个特例需要记住：
1. 未来命题真值会水落石出的我们认为是命题 • 10年之后你一定是千万富翁。 • 火星上有生命 2. 悖论 • 我正在说谎 • 我专门为不为自己理发的人理发
2018/10/11
6
Propositional Variable
• 2+2=5 结论：疑问句、感叹句和祈使句都不是命题
2018/10/11 4
More Examples of Proposition
• 3-x=5.
– x is a variable, so the truth value of this sentence is open.
• 这盘菜很咸。
• 这个人个很高。 结论：聪明的你如果给不出真值那就不是命题
2018/10/11 8
Negation
p(also denoted by p
: it is not the case that p