Consider the following “proof" 1: a ^ (¬a → b) (Premise) 2: a ^ (¬a V b) (Implication, 1) 3: (a ^ ¬a) V b (Associative, 2) 4: T V b (Negation, 3) 5: T (Identity, 4) (a) Identify all of the errors in the proof. (b) Using Rules of Inference and Propositional Equivalences, formally prove that a ^ (¬a → b) = a

Consider the following “proof" 1: a ^ (¬a → b) (Premise) 2: a ^ (¬a V b) (Implication, 1) 3: (a ^ ¬a) V b (Associative, 2) 4: T V b (Negation, 3) 5: T (Identity, 4) (a) Identify all of the errors in the proof. (b) Using Rules of Inference and Propositional Equivalences, formally prove that a ^ (¬a → b) = a

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(x) and Q(x) be relations, with x a variable from a given domain of discourse U. The Axiom of the Hilbert deductive system for first-order logic applied below is "Axiom 1,2,3,4,5" ? ⊢ ∀x P(x) → (∃x Q(x) → ∀x P(x)) Consider the steps given below of a proof in the Hilbert deductive system and determine the reason that justifies each. ∀x A(x) ⊢ ∀x A(x) "?"∀x A(x) ⊢ ∀x A(x) → A(a) " ?"∀x A(x) ⊢ A(a) " ?"
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Simplify the given proposition using Logical Equivalence rules and determine if it is a tautology, contradiction orcontingency. Do not forget to indicate rule/s applied in each step. 1. ¬(q ⊕ p) → ¬(p ∧ q) 2. (¬p ∧ r) ⊕ (p ∨ q)
Consider the wffs:φ1 ≡ p1 → (p2 → (p3 → p4))φ2 ≡ (p1 ∧ p2 ∧ p3) → p4(a) Technically speaking, neither φ1 nor φ2 is well-formed since neither is allowed by the formal syntaxof propositional logic. Correct them. Note, however, that we will freely make such trivial ’errors’ throughout this semester (as do most such courses).(b) Use truth tables (in the form defined in this course) to show that φ1 ↔ φ2.(c) After internalizing an intuitive understanding of this equality, propose an extension of it to n atoms.(d) State the number of rows in a truth table for proving the extension
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2) (L2) Prove using laws of logic that the conditional proposition (p ∧ q) → r is equivalent to (p ∧ ¬ r) →¬ q. 3) (L3) Show that the converse of a conditional proposition p: q → r is equivalent to the inverse of proposition p using a truth table. 4.1) (L4) Show whether ((p ∧ (p→q)) ↔ ¬p) is a tautology or not. Use a truth table and be specific about which row(s)/column(s) of the truth table justify your answer. 4.2) (L4) Give truth values for the propositional variables that cause the two expressions to have different truth values. For example, given p ∨ q and p ⊕ q, the correct answer would be p = q = T, because when p and q are both true, p ∨ q is true but p ⊕ q is false. Note that there may be more than one correct answer. r ∧ (p ∨ q) (r ∧ p) ∨ q
Suppose propositional sentences and knowledge-base here. We have a knowledge-base called KB. KBPrime is defined as an union of (1) KB and (2) the negation of a propositional predicate P. Answer true/false to the following questions. Note that "|=" is used to denote entailment, and "|-" is used to denote derivation through resolution. a) If KB |= P, then KB |- P. That is, if KB entails P, then KB derives (using resolution) P. b) If KB |- P, then KB |= P. c) If KBPrime |- [], then KBPrime is not satisfiable. d) If KBPrime is not satisfiable, then KBPrime |- [].
Using the well-known logical laws, e.g., Identity, Domination, Idempotent, Negation, Double Negation, Commutative, Associative, Distributive, De-Morgan and Absorption Show that the logical expression:q ^ (p → q) Is equivalent to?(qvp)