Section 1.1 Statement, Symbolic Representation, and Tautologies Logic - Propositional Logic Proposition Truth Table Operators(not, ∧, ∨, implication) Proof Technique - Predicate Logic Predicate Quantifier Proof • Def: A proposition(or statement) is a sentence that is either true or false. e.g. Ten is less than seven. (true) YES There are life forms on other planets in the universe. (true or false, but we don’t need to be able to decide) YES e.g. Pass me the butter! (A command) NO How are you? (A question) NO e.g. This rose is white. YES Triangles have 4 vertices. YES 3+2=4 YES Today is my birthday. YES • Propositions are represented by capital letters, e.g.: A, B,…, Z. e.g. P = work is better than vocation. Propositional variable • Def: The truth value of a propositional variable P is true (T) if the proposition assigned to it is true, and false (F) otherwise. The truth(T) or falsity(F) of a proposition is called its truth value. 1 • Logic Operators: Name Symbol Operation Not (unary operator) ′ (¬) e.g. P′,¬P negation And (binary operator) ∧ e.g. P∧Q conjunction or ∨ e.g. P∨Q disjunction implies → e.g. P→Q implication if and only if ↔ e.g. P↔Q equivalence (iff) • Truth tables show the truth value of the result after combining propositional variable(s) with logical operator(s). e.g. NOT AND OR P P Q P Q P′ P ∧ Q P ∨ Q T F T T T T T T F T T F F T F T F T F F T T F F F F F F e.g. implication P → Q is read “If P then Q” “P implies Q” ∴P is antecedent or hypothesis Q is consequent or conclusion. Rule: In logic, the truth value of P→ Q is true if P is false or if Q is true. P Q P → Q T T T T F F F T T F F T e.g. If the rain continues, then the river will flood. 2 e.g. iff P ↔ Q ≡ (P→ Q) ∧ (Q→ P) P Q P → Q Q→ P (P→ Q) ∧ (Q→ P) T T T T T T F F T F F T T F F F F T T T ∴only if both P and Q are T or both are F, the whole statement will be false. Note: How many rows in a truth table with n variables? Answer: 2n • Def: A well-formed formula (wff) is defined as follows: (1) All propositional variables and the constants T & F are wffs. (2) If α and β are wffs, then α′, β′, α∧β, α∨β, α→β, α↔β are wffs. Nothing else. e.g. ( (P∧Q)R′ ) is not a wff. [ ((P∧Q) ∨ (R∧S)) ∧ R′] is a wff, but is different from [(P∧Q) ∨ (R∧S) ∧ R′] • Priority: Pr(not) > Pr(and) > Pr(or) > Pr(→) > Pr(↔) e.g. P = Today is Monday. Q = I’ll go to London. (i) If today is Monday then I won’t go to London. P → Q′ (ii) Today is Monday or I’ll go to London, but not both. P ⊕ Q = ( P′ ∧ Q ) ∨ ( P ∧ Q′ ) = ( P ∨ Q ) ∧ ( P → Q′ ) = P′ ↔ Q (iii) I’ll go to London and today is not Monday. Q ∧ P′ (iv) If and only if today is not Monday then I’ll go to London. P′ ↔ Q 3 • Translate English statements into wffs: (1) If you go bankrupt, you either go to jail or you become a slave. B = you go bankrupt J = you go to jail S = you become a slave B → (J∨S) (2) If you are not good kids, then I will give you hard exams. G = you are good kids H = I will give you hard exams G′ → H (3) If I miss the train today, then I can arrive only 5 minutes late, assuming that the next train is on time. M = I miss the train today. A = I can arrive only 5 min. late T = the next train is on time (M ∧ T) → A or M → ( T → A) Ex: P = Mathematicians are generous. Q = Spiders hate algebra. • Write compound propositions symbolized by: (i) P ∨ Q′ Mathematicians are generous or Spiders don’t hate algebra (or both). (ii) Q ∧ P or ¬( Q ∧ P) It’s not the case that spiders hate algebra and mathematicians are generous. (iii) P → Q If mathematicians are generous then spiders hate algebra. (iv) P′ ↔ Q′ Mathematicians are not generous if and only if spiders don’t hate algebra. 4 • Def: A wff α, whose truth values are always true, is called a tautology. A wff α, whose truth values are always false, is called a contradiction. e.g. P P′ α = P ∨ P′ T F T tautology F T T e.g. P P′ α = P ∧ P′ contradiction T F F F T F • Use truth table to prove that a wff is a tautology or not: e.g. α = (P′ ∨ Q) ↔ (P → Q) P Q P′ P′ ∨ Q P → Q P′ ∨ Q ↔ P → Q T T F T T T T F F F F T F T T T T T F F T T T T ∴α is a tautology. 5 • Some tautology equivalences: (1) commutative properties: A ∨ B ⇔ B ∨ A A ∧ B ⇔ B ∧ A (2) associative properties: (A ∨ B) ∨ C ⇔ A ∨ (B ∨ C) (A ∧ B) ∧ C ⇔ A ∧(B ∧C) (3) distributive properties: A ∨ (B ∧C) ⇔ (A ∨ B) ∧ (A ∨ C) A ∧ (B ∨C) ⇔ (A ∧ B) ∨ (A ∧ C) (4) identity properties: A ∨ F ⇔ A A ∧ T ⇔ A (5) complement properties: A ∨ A′ ⇔ T A ∧ A′ ⇔ F (6) De Morgan’s Law: (A ∨ B)′ ⇔ A′ ∧ B′ (A ∧ B)′ ⇔ A′ ∨ B′ (7) Double negative: (A′)′ ⇔ A (8) Rewriting implication: (A → B) ⇔ A′ ∨ B (9) Contraposition: (A→ B) ⇔ (B′ → A′) (10) Conditional proof: A → (B → C) ⇔ (A ∧ B) → C 6.