Decide which of the following formulas are valid. Justify your answer by giving a proof (incase of valid formulas) or by proposing a falsifying model (in case of invalid formulas)(a) Vx (P(x) → Q(x)) → (3x P(x)→ 3x Q(x))(b) 3x (P(x) → Q(x)) → (Vx P(x)→ Vx Q(x))(c) 3x (P(x)→ Q(x)) → (3x P(x)→ 3x Q(x))(d) 3x (P(x) → Q(x)) → (Vx P(x)→ 3x Q(x))Which of the sentences are satisfiable. Prove your answers.

Answered: Image /qna-images/answer/77f3509b-d6ec-45a7-81ac-ec87c64153c6.jpg

Answered: (a) Vx (P(x)→ Q(x)) → (3x P(x)→→ 3x…

(a) Vx (P(x)→ Q(x)) → (3x P(x)→→ 3x Q(x)) ((x)0 XA (x)d XA) ((x)0 (x)d) XE (9) (c) 3x (P(x) → Q(x)) → (3x P(x)→ 3x Q(x)) ((x)O XE + (x)d XA) + ((x)O− (x)d) XE (P)

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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Question

Decide which of the following formulas are valid. Justify your answer by giving a proof (in
case of valid formulas) or by proposing a falsifying model (in case of invalid formulas)
(a) Vx (P(x) → Q(x)) → (3x P(x)→ 3x Q(x))
(b) 3x (P(x) → Q(x)) → (Vx P(x)→ Vx Q(x))
(c) 3x (P(x)→ Q(x)) → (3x P(x)→ 3x Q(x))
(d) 3x (P(x) → Q(x)) → (Vx P(x)→ 3x Q(x))
Which of the sentences are satisfiable. Prove your answers.

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