If A, B, and C are Boolean variables, which of the following statements are correct? a) A ∧ (B ∨ C) = (A ∧ B) ∨ (A ∧ C) b) A ∨ (B ∧ C) = (A ∨ B) ∧ (A ∨ C) c) (A ∧ B) ∨ C = C ∨ (B ∧ A)
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If A, B, and C are Boolean variables, which of the following statements are correct?
a) A ∧ (B ∨ C) = (A ∧ B) ∨ (A ∧ C)
b) A ∨ (B ∧ C) = (A ∨ B) ∧ (A ∨ C)
c) (A ∧ B) ∨ C = C ∨ (B ∧ A)
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- Construct formal proofs for the following arguments: 1. ¬(A ∨ B) ⊢ ¬A 2. A, ¬B ⊢ ¬(A → B) 3. (A ∧ B) ∨ (A ∧ C) ⊢ B ∨ C 4. A ⊢ B → (A ∧ B)Construct proof for the following argument within the system of sentential logic: 1. ~(~D ⊃ ~C) ⊃ ~B Premise2. ~B ⊃ A Premise3. (Y V C) & (~C V ~A) Premise /: . D V (A V Y)What should the pre-condition P for the statement below to be an instance of Hoare’s axiom scheme? All variables are of type int. P { x = y + z; } ForAll(z = 1; z < 100) x + 2*z > w + 2
- 1. In transforming a formula into an equivalent CNF, you can use the absorption laws to eliminate conjunctions within disjuntions, i.e. expressions such as p ˅ (q ˄ r) and (p ˄ q) ˅ r. True False 2. Let p, q, and r be propositional variables. Which of the following expressions would NOT be formulas in conjunctive normal form? (p ˄ ¬q) ˅ (¬r ˄ q ˄ p) ¬q p ˅ q ˅ ¬p p ˄ ¬p 3. Consider the propositional logic formula (p ˄ q) ˅ (r ˅ ¬s) From the options below, which one is the equivalent CNF? To determine the correct answer, transform the formula above into CNF using the steps learnt in this module. (p ˅ r ˅ s) ˄ (q ˅ r ˅ ¬s) (¬p ∨ r ∨ s) ∧ (¬q ∨ ¬r ∨ ¬s) (p ˅ r ˅ ¬s) ˄ (q ˅ r ˅ s) (p ˅ r ˅ ¬s) ˄ (q ˅ r ˅ ¬s)…Determine the truth value of the propositional form below given that p is false, q is true, and r is true. (Identify if it is True or False): r∨(p∧∼q)Use propositional logic to prove that the following arguments are valid: (a) (A→¬C) ∧(B→A) →(C→(¬A∧¬B)) (b) (A→¬C) ∧(B∨C) ∧¬B→¬A (c) (A→(B→C)) →(B→(A→C)) (d) ¬A∧(¬B∨C) →((A∨B) →(¬A∧C))
- if p and q are logical variables, which of the following is a tautology (i.e., always correct irrespective of specific value of variables) Select one: a. p → (q ∧ p) b. p ∨ (q → q) c. (p ∨ q) → q d. p ∨ (p → q)Use propositional logic to prove that the argument is valid. (A→(B ∨ C))∧¬C→(A→B)Answer the given question with a proper explanation and step-by-step solution. An else statement is required for every if statement. True False
- Write the following English statements using the following predicates and any needed quantifier. The domain of all variables are all people associated in a university S(x): x is a student F(x): is a faculty member A(x, y): x has asked y a question There are at least two students who have asked every faculty member a question There is a faculty member who has asked every other faculty member a questionc) Express each of these statements using quantifiers. Form the negation of each: (i) Some old dogs can learn new tricks. (ii) No rabbit knows calculus. (iii) Every bird can fly. (iv) There is no dog that can talk (v) There is no one in this class who knows French and RussianThe boolean operation implication is defined by the following truth table true true true true false false false false false false true false define a lambda calculus representation for implication: def implies = λx.λy… Show the definition satisfies the truth table for all boolean values of x and y using the definition of implies ((implies false) false)-> true ((implies false) true)-> true ((implies true) false)-> false ((implies true) true)-> true