An Introduction to Logic 1.1 ~ 1.4 6/21/04 ~ 6/23/04
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1 An Introduction to Logic 1.1 ~ 1.4 6/21/04 ~ 6/23/04 1
2 A Taste of Logic Logic puzzles (1) Knights and Knaves Knights: always tell the truth Knaves: always lie You encounter two people A and B. A says: "B is a knight" B says: "The two of us are opposite types" Q: What are A and B? (2) Murder suspects 3 suspects for murdering Cooper: Smith, Jones, Williams Smith: "Cooper is a friend of Jones, and Williams dislikes him" Jones: "I don't know Cooper and I am out of town that day" Williams:"I saw both Smith and Jones with Cooper that day, and either Smith or Jones must have killed him" Q: Who is the murder? 2
3 What is Logic? Mathematical Logic is a tool for providing precise meaning to mathematical statements. It includes: A formal language for expressing them. A concise notation for writing them. A methodology for objectively reasoning about their truth or falsity. Applications of Logic It is the foundation for expressing formal proofs in all branches of mathematics. In computer science Circuit design Program constructing Program verification 3
4 1.1 Logic DEF1: A proposition is a declarative sentence that is either true (T) or false (F), but not both. Ex1: propositions 1) Beijing is the capital of China 2) = 5 3) = 4 Ex2: not propositions 1) Is your major computer science? (question) 2) Finish the homework by yourself. (command) 3) What a smart student! (exclamation) 4) x + 1 = 3 (could be both T and F) Exercise1: proposition or not? 1) You are a student. 2) Lingli is a cute girl. 3) There is life on Mar. 4) x + 1 = 5 if x = 1 5)
5 Logical operators DEF2: A proposition variable is a variable such as p, q, r over the boolean domain {T,F} DEF3: A logical operator is a rule defined by a truth table Some popular logical operators Operator Negation Conjunction Disjunction Exclusive-OR Implication Biconditional Nickname NOT AND OR XOR IMPLIES IFF notation 5
6 The Negation Operator Expression: "It is not the case that p" "p is false" Notation: p Truth table: Ex3: p p T F F T p = "Today is Friday" p = "Today is not Friday", or "It is not the case that today is Friday 6
7 The Conjunction Operator Expression: "p and q" "Both p and q are true" Notation: p q Truth table: p q p q F F F F T F T F F T T T Ex4: p = "Today is Friday" q = "It is raining today" p q = "Today is Friday and it is raining today" 7
8 The Disjunction Operator Expression "p or q" "Either p or q, or possible both are true" Notation: p q Truth table: p q p q F F F F T T T F T T T T Ex5: p = "Today is Friday" q = "It is raining today" p q = "Today is Friday or it is raining today" Note: Disjunction is also called "inclusive or" 8
9 The Exclusive OR Operator Expression: "Either p or q is true, but not both are true" Notation: p q Truth table: Ex6 p = "I will marry Helen" p q p q F F F F T T T F T T T F q = "You will marry Helen" p q = "Either you or I will marry Helen" 9
10 English "or" is ambiguous Ex7: "Tomorrow will be sunny or partially cloudy" Inclusive? Exclusive? Ex8: "Joe's major is computer science or chemical engineering" Inclusive? Exclusive? Note: We can infer the meaning of English "or" based on context, but not always Logic is more precise than natural languages Convention: In this class, "or" in default is inclusive 10
11 The Implication operator Expression: "If p then q", "if p, q", "p implies q", "q if p", "q whenever p", "q follows from p" "q is necessary for p", "p is sufficient for q", p q p q F F T F T T T F F T T T "p only if q", Notation: p q Truth table: Note: p q is F only when p is true and q is false. q can be true even if p is false p q does not require that p or q is true 11
12 The Implication operator Ex9: p = "Today is Friday" q = "2 + 3 =6" p q = "If today is Friday, then 2+3=6", T or F? Ex10: "1=0" "pigs can fly", T or F? "1=1" "pigs can fly", T or F? Related implications of p q: Negation: (p q) Converse: q p Contrapositive: q p Inverse: p q Note: only the contrapositive of p q has the same truth value as p q 12
13 The Biconditional Operator Expression: "p if and only if q" "p is necessary and sufficient for q" Notation: p q Truth table: Note: Ex11 this truth table is exactly the opposite of 's p q (p q) p = "You can take the flight" p q p q F F T F T F T F F T T T q = "You buy a ticket" p q = "You can take a flight if and only of you buy a ticket 13
14 Precedence of Logical Operators Ex12: r s q ( (r s)) q? ( r) (s q)? (r (s q))? (( r) s) q? Precedence Order from high to low:,,,,, and are left associative: p q r (p q) r p q r (p q) r 14
15 Logical Operators Operator Nickname Notation Precedence Negation NOT p 1 Conjunction AND p q 2 Disjunction OR p q 3 Exclusive-OR XOR p q 4 Implication IMPLIES p q 5 Biconditional IFF p q 6 15
16 Compound Propositions DEF4: An atomic propositional form is either a boolean constant or a propositional variable. DEF5: An compound propositional form is constructed by combining atomic propositional forms using logical operators Ex13: Some compound propositional forms on two variables: p, p q, p q, p q, p q, p q, (p q) q, Ex14: How can this English sentence be translated into a logical expression? "You can access the Internet from campus only if you are a computer science major or you are not a freshman." 16
17 Analyze compound propositions Ex15: Q: Analyze "p (r q)" using truth table A: p q r q r q (r q) p (r q) F F F T T F F F F T T T F F F T F F T F F F T T F F T F T F F T T F F T F T T T F F T T F F T F F T T T F F T T 17
18 Logic and Bit Operations Bit: 0 or 1 Represent truth values using bit: 1--T, 0--F Bit operators correspond to the logical operators by replacing T by 1 and F by 0 Bit operations can be extended to operate on bit strings: (bit-wise) Bit-wise OR Bit-wise AND Bit-wise XOR 18
19 Logical Puzzles Ex16: Knights and Knaves Knights: always tell the truth Knaves: always lie You encounter two people A and B. A says: "B is a knight" B says: "The two of us are opposite types" Q: What are A and B? 19
20 Logical Puzzles Exercise2: Murder suspects 3 suspects for murdering Mr. Cooper: Smith, Jones, Williams. All of them denied the murdering. Smith said: "Cooper is a friend of Jones, and Williams dislikes Cooper." Jones said: "I don't know Cooper and I am out of town that day." Williams said:"i saw both Smith and Jones with Cooper that day, and either Smith or Jones must have killed him" Q: Who is the murder? 20
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