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Problems: Use the truth table method to solve the following problems: 4. Propositional Logic Using truth tables 1. Decide whether the formula p0→¬p0 is a tautology, a contingency, or a contradiction. 2. Decide whether the formula p0v¬(p0∧p1) is a tautology, a contingency, or a contradiction. Problems Jouko Väänänen: Propositional logic Problems: Use the truth table method to solve the following problems: 1. Decide whether p0→p1 is equivalent to ¬(p1→p0) or not. 2. Decide whether ¬p0 ∨p1 is equivalent to ¬(p0 ∧p1) or not. Last viewed Problems: Use the truth table method to prove the equivalence of 1. ¬(A ∧ B) and ¬A ∨ ¬B 2. ¬¬A and A 3. A∧A and A 4. A∨A and A Jouko Väänänen: Propositional logic Last viewed Problems: Use the truth table method to prove the equivalence of 1. A→B and ¬A ∨ B 2. A!B and (A→B)∧(B→A) Jouko Väänänen: Propositional logic Last viewed Jouko Väänänen: Propositional logic Last viewed Problems: Use the truth table method to prove the equivalence of 1. A∨(B∨C) and (A∨B)∨C 2. A∧(B∧C) and (A∧B)∧C Jouko Väänänen: Propositional logic Last viewed Problems: Use the truth table method to prove the equivalence of Problems: Use the truth table method to prove the equivalence of 1. A∧(B∨C) and (A∧B)∨(A∧C) 1. A→(B→C) and B→(A→C) 2. A∨(B∧C) and (A∨B)∧(A∨C) 2. A→(B→C) and (A∧B)→C Jouko Väänänen: Propositional logic Last viewed Exercise ! Jouko Väänänen: Propositional logic Last viewed Problem (Negation Normal Form) Show that a conjunction of implications pi→pi* is not a tautology if and only if there is an i such that i is not i*. Jouko "äänänen: Propositional logic Define for formulas built up using just ¬, ∧ and ∨, which are in negation normal form: • (pi)*=pi, (¬pi)*=¬pi (A∧B)*=A*∨B* • (A∨B)*=A*∧B* Define for formulas built up using just ¬, ∧ and ∨: ! (pi)+=pi, (pi)-=¬pi ! (¬A)+=A-, (¬A)-=A+ (A∧B)+=A+∧B+, (A∧B)-=A-∨B- ! (A∨B)+=A+∨B+, (A∨B)-=A-∧B- ! A+ is called the Negation Normal Form of A. It has negations only in front of proposition symbols. ! ! Find (¬((p0∧p1)∨p2))+. ! Show that A+ and A are equivalent. ! Show that A- and ¬A are equivalent. Last viewed Problem (Dual of a formula) • ! The formula A* is the dual of A. It is obtained from A by switching conjunctions to disjunctions. 1. What is the dual of (p0∨¬p1)∧p2? 2. Show that if A is a tautology, so is ¬A*. 3. Show that if A and B are equivalent, then so Jouko Väänänen: Propositional logic Last viewed Problem (Substitution) ! Suppose A is a propositional formula in which the proposition symbols p0,…,pn-1 occur. Let Ai be for each i=0,…,n-1 an arbitrary propositional formula. Let A’ be the result of substituting everywhere in A the formula Ai for pi. ! Show that if A is a tautology, then so is A’. are A* and B*. Jouko Väänänen: Propositional logic Last viewed Jouko Väänänen: Propositional logic Last viewed Problem (Topological interpretation) ! A topological interpretation is a topological space E and a function f from proposition symbols to open sets in E. We extend this to propositional formulas built up from ∧, ∨ and ¬: " f(A∧B)=f(A)⋂f(B) " f(A∨B)=f(A)∪f(B) " f(¬A)=Int(E-f(A)), the interior of the complement of f(A) in the topological sense. ! Show that A is a tautology iff f(A)=E in every topological interpretation where E is a discrete space. ! A is said to be a constructive tautology if f(A)=E in every topological interpretation. ! Show that ¬(A∧¬A) is, but A∨¬A is not a constructive tautology. Jouko Väänänen: Propositional logic Last viewed