Translating Sentences into Logical Expressions and Solving Puzzles - Prof. Alper Ungor, Assignments of Computer Science

Download Translating Sentences into Logical Expressions and Solving Puzzles - Prof. Alper Ungor and more Assignments Computer Science in PDF only on Docsity! COT3100: Propositional Equivalences 1 Logic and Proofs Alper Üngör Jan 2007 COT3100: Applications of Discrete Structures Jan 2007 Alper Üngör COT3100: Propositional Equivalences 2 1 – Translating from Natural Languages EXAMPLE. Translate the following sentence into a logical expression: “I met my ex-girlfriend today and either she grew taller or I got shorter.” p = “I met my ex-girlfriend today.” q = “She grew taller.” r = “I got shorter.” p ∧ (q ∨ r) p ∧ q ∨ r would mean something else: “I got shorter or I met my ex-girlfriend today and she grew taller” p ∧ (q ⊕ r) would also mean something else... COT3100: Applications of Discrete Structures Jan 2007 Alper Üngör COT3100: Propositional Equivalences 5 Let p= “Alice is a knight.” ¬p= “Alice is a knave.” Let q= “Bob is a knight.” ¬q= “Bob is a knave.” Alice said q and Bob said p ⊕ q. Island Rule: Alice is a knight iff what she said is true and Bob is a knight iff what he said is true. (p ↔ q) ∧ (q ↔ (p ⊕ q)) Alice said Bob said Island rule p q ¬p ¬q p ⊕ q (p ↔ q) ∧ (q ↔ (p ⊕ q)) T T F F F F T F F T T F F T T F T F F F T T F T Hence, Both Alice and Bob are knaves. COT3100: Applications of Discrete Structures Jan 2007 Alper Üngör COT3100: Propositional Equivalences 6 3 – Tautology and Contradiction A compound statetement that is always true is called a tautology. A compound statetement that is always false is called a contradiction. A compound proposition that is neither a tautology nor a contradiction is called a contingency COT3100: Applications of Discrete Structures Jan 2007 Alper Üngör COT3100: Propositional Equivalences 7 4 – Equivalence The compound propositions p and q are called logically equivalent, denoted by p ≡ q, if p ↔ q is a tautology. Remark: The symbol ≡ is not a logical connective and p ≡ q is not a compound proposition but rather is the statement that p ↔ q is a tautology. Use truth tables to determine logical equivalences. Two compound propositions are equivalent if their columns in the truth table are the same. COT3100: Applications of Discrete Structures Jan 2007 Alper Üngör COT3100: Propositional Equivalences 10 6 – Equivalence Example EXAMPLE. “If you are a freshman you can get a free ticket” “Either you are not a freshman or you can get a free ticket.” COT3100: Applications of Discrete Structures Jan 2007 Alper Üngör COT3100: Propositional Equivalences 11 7 – Distributive Law p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r) p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r) COT3100: Applications of Discrete Structures Jan 2007 Alper Üngör COT3100: Propositional Equivalences 12 Equivalence Name p ∧ T ≡ p Identity Laws p ∨ F ≡ p p ∨ T ≡ T Domination Laws p ∧ F ≡ F p ∨ p ≡ p Idempotent Laws p ∧ p ≡ p ¬(¬p) ≡ p Double Negation Law p ∨ q ≡ q ∨ p Commutative Laws p ∧ q ≡ q ∧ p COT3100: Applications of Discrete Structures Jan 2007 Alper Üngör COT3100: Propositional Equivalences 15 COT3100: Applications of Discrete Structures Jan 2007 Alper Üngör COT3100: Propositional Equivalences 16 8 – Constructing new equivalences EXAMPLE. Show that ¬(p → q) and p ∧ ¬q are logically equivalent. ¬(p → q) ≡ ¬(¬p ∨ q) (See page 10 of these slides) ≡ ¬(¬p) ∧ ¬q (De Morgan’s Law) ≡ p ∧ ¬q (Double negation) COT3100: Applications of Discrete Structures Jan 2007 Alper Üngör COT3100: Propositional Equivalences 17 9 – Proving a Tautology EXAMPLE. Show that (p ∧ q) → (p ∨ q) is a tautology. (p ∧ q) → (p ∨ q) ≡ ¬(p ∧ q) ∨ (p ∨ q) (See page 10 of these slides) ≡ (¬p ∨ ¬q) ∨ (p ∨ q) (De Morgan’s Law) ≡ (¬p ∨ ¬q ∨ p) ∨ q (Associative Law) ≡ (¬p ∨ p ∨ ¬q) ∨ q (Commutative Law) ≡ (¬p ∨ p) ∨ (¬q ∨ q) (Associative Law) ≡ T ∨ T (Negation Law) ≡ T COT3100: Applications of Discrete Structures Jan 2007 Alper Üngör