Applications in Computer ScienceDesign of computer circuits • Construction of computer programs • Verification of the correctness of programs • Constructing proofs automatically • Artificial intelligence
Logics Tran Vinh Tan Chapter Logics Discrete Mathematics I on 13 March 2012 Contents Propositional Logic Tran Vinh Tan Faculty of Computer Science and Engineering University of Technology - VNUHCM 1.1 Logics Contents Tran Vinh Tan Contents Propositional Logic Propositional Logic 1.2 Logics Logic Tran Vinh Tan Definition (Averroes) The tool for distinguishing between the true and the false Contents Propositional Logic Definition (Penguin Encyclopedia) The formal systematic study of the principles of valid inference and correct reasoning Definition (Discrete Mathematics - Rosen) Rules of logic are used to distinguish between valid and invalid mathematical arguments 1.3 Logics Applications in Computer Science Tran Vinh Tan • Design of computer circuits Contents Propositional Logic • Construction of computer programs • Verification of the correctness of programs • Constructing proofs automatically • Artificial intelligence • Many more 1.4 Logics Propositional Logic Tran Vinh Tan Definition A proposition is a declarative sentence that is either true or false, but not both Contents Propositional Logic Examples • Hanoi is the capital of Viet Nam • New York City is the capital of USA • 1+1=2 • 2+2=3 1.5 Logics Examples Tran Vinh Tan Examples (Which of these are propositions?) • How easy is logic! Contents Propositional Logic • Read this carefully • H1 building is in Ho Chi Minh City • 4>2 • 2n ≥ 100 • The sun circles the earth • Today is Thursday • Proposition only when the time is specified 1.6 Logics Notations Tran Vinh Tan Contents Propositional Logic • Propositions are denoted by p, q, • The truth value (”chân trị”) is true (T) or false (F) 1.7 Logics Operators Tran Vinh Tan Negation - ”Phủ định”: ¬p Contents Propositional Logic Bảng: Truth Table for Negation p ¬p T F F T 1.8 Logics Operators Tran Vinh Tan Conjunction - ”Hội”: p ∧ q “p and q” Disjunction - ”Tuyển”: p ∨ q “p or q” Contents Propositional Logic p q p∧q p q p∨q T T F F T F T F T F F F T T F F T F T F T T T F I’m teaching DM1 and it is raining today We need students who have experience in Java or C++ Tomorrow, I will eat Pho or Bun bo 1.9 Logics Operators Tran Vinh Tan Exclusive OR - Tuyển loại: p ⊕ q “p or q (but not both)” Implication - Kéo theo: p → q “if p, then q” p q p⊕q p q p→q T T F F T F T F F T T F T T F F T F T F T F T T Contents Propositional Logic If it rains, the pavement will be wet 1.10 Logics More Expressions for Implication p → q Tran Vinh Tan • if p, then q • p implies q Contents Propositional Logic • p is sufficient for q • q if p • p only if q ã q unless ơp ã If you get 100% on the final, you will get 10 grade • If you feel asleep this afternoon, then + = 1.11 Logics Conditional Statements From p → q Tran Vinh Tan Contents Propositional Logic • q → p (converse - o) ã ơq ơp (contrapositive - phản đảo) • Prove that only contrapositive have the same truth table with p→q 1.12 Logics Tran Vinh Tan Exercise What are the converse and contrapositive of the following conditional statement “If he plays online games too much, his girlfriend leaves him.” Contents Propositional Logic • Converse: If his girlfriend leaves him, then he plays online games too much • Contrapositive: If his girlfriend does not leave him, then he does not play online games too much 1.13 Logics Biconditionals Tran Vinh Tan p↔q “p if and only if q” Contents p q p→q T T F F T F T F T F F T Propositional Logic • “p is necessary and sufficient for q” • “if p then q, and conversely” • “p iff q” 1.14 Logics Translating Natural Sentences Tran Vinh Tan Exercise I will buy a new phone only if I have enough money to buy iPhone or my phone is not working Contents Propositional Logic • p: I will buy a new phone • q: I have enough money to buy iPhone • r: My phone is working • p → (q ∨ ¬r) 1.15 Logics Translating Natural Sentences Tran Vinh Tan Contents Propositional Logic Exercise He will not run the red light if he sees the police unless he is too risky 1.16 Logics Construct Truth Table Tran Vinh Tan Exercise Construct the truth table of the compound proposition (p ∨ ¬q) → (p ∧ q) p q ¬q p ∨ ¬q p∧q (p ∨ ¬q) → (p ∧ q) T T F F T F T F F T F T T T F T T F F F T F T F Contents Propositional Logic 1.17 Logics Applications Tran Vinh Tan Contents • System specifications • “When a user clicked on Help button, a pop-up will be shown up” Propositional Logic • Boolean search • type “dai hoc bach khoa” in Google • means “dai AND hoc AND bach AND khoa” 1.18 Logics Applications (cont.) Tran Vinh Tan • Logic puzzles • There are two kinds of inhabitants on an island, knights, who always tell the truth, and their opposites, knaves, who always lie You encounter two people A and B What are A and B if A says “B is a knight” and B says ”The two of us are opposite types”? Contents Propositional Logic • Bit operations • 101010011 is a bit string of length nine 1.19 Logics Tautology and Contradiction Tran Vinh Tan Definition A compound proposition that is always true (false) is called a tautology (contradiction) Contents Propositional Logic • Tautology: • Contradiction: mâu thun Example ã p ơp (tautology) ã p ¬p (contradiction) 1.20 Logics Logical Equivalences Tran Vinh Tan Contents Definition Propositional Logic The compound compositions p and q are called logically equivalent if p ↔ q is a tautology, denoted p ≡ q Example Show that ¬(p ∨ q) and ¬p ∧ ¬q are logically equivalent 1.21 Logics Logical Equivalences Tran Vinh Tan p∧T p∨F ≡ ≡ p p Identity laws Luật đồng p∨T p∧F ≡ ≡ T F Domination laws Luật nuốt p∨p p∧p ≡ ≡ p p Idempotent laws Luật lũy đẳng ¬(¬p) ≡ p Double negation law Luât phủ định kép Contents Propositional Logic 1.22 Logics Logical Equivalences Tran Vinh Tan p∨q p∧q ≡ ≡ q∨p q∧p (p ∨ q) ∨ r (p ∧ q) ∧ r ≡ ≡ p ∨ (q ∨ r) p ∧ (q ∧ r) Associative laws Luật kết hợp p ∨ (q ∧ r) p ∧ (q ∨ r) ≡ ≡ (p ∨ q) ∧ (p ∨ r) (p ∧ q) ∨ (p ∧ r) Distributive laws Luật phân phối ¬(p ∧ q) ¬(p ∨ q) ≡ ≡ ¬p ∨ ¬q ¬p ∧ ¬q De Morgan’s law Luật De Morgan p ∨ (p ∧ q) p ∧ (p ∨ q) ≡ ≡ p p Commutative laws Luật giao hoán Contents Propositional Logic Absorption laws Luật hút thu 1.23 Logics Logical Equivalences Tran Vinh Tan Equivalence p ∨ ¬p p ∧ ¬p (p → q) ∧ (p → r) (p → r) ∧ (q → r) (p → q) ∨ (p → r) (p → r) ∨ (q → r) p↔q Contents ≡ ≡ ≡ ≡ ≡ ≡ ≡ Propositional Logic T F p → (q ∧ r) (p ∨ q) → r p → (q ∨ r) (p ∧ q) → r (p → q) ∧ (q → p) 1.24 Logics Constructing New Logical Equivalences Tran Vinh Tan Example Show that ¬(p ∨ (¬p ∧ q)) and ¬p ∧ ¬q are logically equivalent by developing a series of logical equivalences Contents Solution ¬(p ∨ (¬p ∧ q)) Propositional Logic ≡ ¬p ∧ ¬(¬p ∧ q) by the second De Morgan law ≡ ¬p ∧ [¬(¬p) ∨ ¬q] by the first De Morgan law ≡ ¬p ∧ (p ∨ ¬q) by the double negation law ≡ (¬p ∧ p) ∨ (¬p ∧ ¬q) by the second distributive law ≡ F ∨ (¬p ∧ ¬q) because ¬p ∧ p ≡ F ≡ ¬p ∧ ¬q by the identity law for F Consequently, ¬(p ∨ (¬p ∧ q)) and ¬p ∧ ¬q are logically equivalent 1.25