1. Decide whether each of the following sentences is valid, satisfiable or unsatisfiable. a) Smoke Fire b) Smoke-Smoke c) Smoke v Fire v-Fire d) (Smoke Fire) → (→Smoke →→Fire) 2. Given the following sentences in propositional calculus: 1. (ru) 2. (u →→w) 3. (rw) Use resolution refutation to prove →w. Joe doesn't get wet 3. Represent the following sentences in first-order logic, using a consistent vocabulary that you define: If it rains, Joe brings his umbrella If Joe has an umbrella, he doesn't get wet If it doesn't rain, Joe doesn't get wet a) Some students took AI in Spring 2022. b) Every student who takes AI passes it. c) Only one student took AI in Spring 2022. d) No one took AI in Fall 2021.

1. Decide whether each of the following sentences is valid, satisfiable or unsatisfiable. a) Smoke Fire b) Smoke-Smoke c) Smoke v Fire v-Fire d) (Smoke Fire) → (→Smoke →→Fire) 2. Given the following sentences in propositional calculus: 1. (ru) 2. (u →→w) 3. (rw) Use resolution refutation to prove →w. Joe doesn't get wet 3. Represent the following sentences in first-order logic, using a consistent vocabulary that you define: If it rains, Joe brings his umbrella If Joe has an umbrella, he doesn't get wet If it doesn't rain, Joe doesn't get wet a) Some students took AI in Spring 2022. b) Every student who takes AI passes it. c) Only one student took AI in Spring 2022. d) No one took AI in Fall 2021.

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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Let p, q and r be propositions: p: Jasper has sore eyes q: Jasper misses the last day of the board exam r: Jasper passes the board exam Write the following propositions in terms of the given connectives: a.) q→¬r b.) q ↔ p c.) p → q ⋀ ¬r
(a) Let p, q, and r be the propositionsp : You study hard.q : You are on meritr : You do not get degree from Iqra University.Express each of these propositions as an English sentence.(i) ¬q ↔ r (ii) q →¬ r (iii) p∨q ∨¬r (iv) (p →¬r) ∨ (q →¬r
3. Prove ⊢ (¬A → A) → A in Hilbert deductive system. Note: In addition to the axioms and rule of inference of H, you may use any of the derived rules and/or theorems 3.20-3.30 (as numbered in the textbook). You may not use theorem 3.31, as this is precisely that theorem.Theorem 3.20 ⊢ ¬A → (A → B). Theorem 3.21 ⊢ A → (¬A → B). Theorem 3.22 ⊢ ¬¬A → A. Theorem3.23 ⊢A→¬¬A.Theorem3.25 ⊢(A→B)→(¬B→¬A).Theorem3.28 ⊢(¬A→false)→A.Rule 3.29 (Reductio ad absurdum) Theorem3.30 ⊢(A→¬A)→¬A. Prove {¬A} ⊢ (¬B → A) → B in H. Note: In addition to the axioms and rule of inference of Hilbert deductive system, you may use any of the derived rules and/or theorems 3.20-3.30 (as numbered in the textbook).
Suppose you are given some facts in First-Order Logics:a) Andi is a professorb) All professors are people.c) Ani is the dean.d) All Deans are professors.e) All professors consider the dean a friend or don’t know him.f) Everyone is a friend of someone.g) People only criticize people that are not their friends. h) Andi criticized Ani.Prove that: Ani is not Andi’s friend.
1): Select the law which shows that the two propositions are logically equivalent. ¬((w∨p)∧(¬q∧¬w)) ¬(w∨p)∨¬(¬q∧¬w) A): DeMorgan’s law B): Distributive law C): Associative law D): Complement law 2): Select the proposition that is a tautology. (p∧q)→¬p (p∨q)→p (p∧q)↔p 4 . (p∧q)→p A): 4 B): 1 C): 2 D): 3
) Show that ∀xP(x) ∧ ∃xQ(x) is logically equivalent to ∀x∃y(P(x) ∧ P(y)) The quantifiers have the same non empty domain. I know that to prove a proposition is logically equivalent to another one, I have to show that ∀xP(x) ∧ ∃xQ(x) ↔ ∀x∃y(P(x) ∧ P(y)) Which means I have to prove that (∀xP(x) ∧ ∃xQ(x)) → ∀x∃y(P(x) ∧ P(y)) ∧ ∀x∃y(P(x) ∧ P(y)) → (∀xP(x) ∧ ∃xQ(x)) I don't know the answer, so I saw the textbook answer. It says (1) Suppose that ∀xP(x) ∧ ∃xQ(x) is true. Then P(x) is true for all x and there is an element y for which Q(y) is true. I get this part. Because P(x) ∧ Q(x) is true for all x and there is a y for which Q(y) is true, ∀x∃y(P(x) ∧ P(y)) is true. Emm... I think ∀x∃y(P(x) ∧ P(y)) is true because ∀x only affects P(x) and ∃y only affects P(y) since their alphabets are different. So, it has the exact same meaning as ∀xP(x) ∧ ∃yQ(y). And since the domains are the same, ∀xP(x) ∧ ∃yQ(y) is actually equal to ∀xP(x) ∧ ∃xQ(x). But the textbook states that "P(x) ∧ Q(x) is…
1. Construct a detailed truth table for the following propositional statement: (((P ∨Q)∧¬P)→Q) Is it a tautology? 2. Construct a detailed truth table for the following propositional statement: (P ∧ Q) → R Is it a tautology?
The following two sentences of TFL are not equivalent. (R → ¬ T); (¬ R ∧ T) (a) Determine what method—truth table or proof—you would need to use to show that the sentences are not equivalent. Then state (in a sentence) how to use that method to show the sentences are not equivalent. (b) Use the method you described in part (a) to show that the sentences are not equivalent.
Proposition (Distributive Law): For expressions P1, P2, P3, any word matching the regular expression (P1 (P2|P3)) also matches the regular expression ((P1P2) (P1P3)) Give a proof of the above proposition, or demonstrate that it is false.
4. You are given the following sentences in propositional logic. Convert them to CNF and prove that "C A - D" is entailed by the sentences using resolution with refutation. Show the steps in the resolution proof. 1. - A = (B v C) 2. A = B 3. - (- В %3D D) → D)
Q4. Letr and y be propositional variables. Which of the following propositions is not a tautology? A. x→ I B. (x→y) V (y →x) C. -(x^-x) D. y → (TV-x) E. x→→x F. (y^-y) → x
Determine whether the following proposition is a tautology.