Use De Morgan's law for quantified statements and the laws of propositional logic to show the following equivalences: **Transcription for Educational Website:****Logical Equivalence in Predicate Logic**In the study of logic, particularly predicate logic, we often deal with statements involving quantifiers and logical connectives. Below is an expression demonstrating logical equivalence:(b) \( \neg \forall x (\neg P(x) \rightarrow Q(x)) \equiv \exists x (\neg P(x) \land \neg Q(x)) \)**Explanation:**This expression is a logical equivalence that shows the transformation between two statements:- The left side, \( \neg \forall x (\neg P(x) \rightarrow Q(x)) \), represents the negation of a universal quantification. It reads as "It is not true that for all \( x \), if \( P(x) \) is false, then \( Q(x) \) is true."- The right side, \( \exists x (\neg P(x) \land \neg Q(x)) \), represents an existential quantification. It reads as "There exists an \( x \) such that \( P(x) \) is false and \( Q(x) \) is false."The equivalence illustrates how negating a universal statement can be converted into an existential statement with different conditions. This concept is fundamental in simplifying and transforming logical expressions within proofs and logical reasoning.

The negation of ∀x.P is ∃x.(¬P) Similarly, ¬(∃x.P)≡∀x(.¬P)

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Use De Morgan's law for quantified statements and the laws of propositional logic to show the following equivalences:

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

Use De Morgan's law for quantified statements and the laws of propositional logic to show the following equivalences:

**Transcription for Educational Website:**

**Logical Equivalence in Predicate Logic**

In the study of logic, particularly predicate logic, we often deal with statements involving quantifiers and logical connectives. Below is an expression demonstrating logical equivalence:

(b) \( \neg \forall x (\neg P(x) \rightarrow Q(x)) \equiv \exists x (\neg P(x) \land \neg Q(x)) \)

**Explanation:**

This expression is a logical equivalence that shows the transformation between two statements:

- The left side, \( \neg \forall x (\neg P(x) \rightarrow Q(x)) \), represents the negation of a universal quantification. It reads as "It is not true that for all \( x \), if \( P(x) \) is false, then \( Q(x) \) is true."

- The right side, \( \exists x (\neg P(x) \land \neg Q(x)) \), represents an existential quantification. It reads as "There exists an \( x \) such that \( P(x) \) is false and \( Q(x) \) is false."

The equivalence illustrates how negating a universal statement can be converted into an existential statement with different conditions. This concept is fundamental in simplifying and transforming logical expressions within proofs and logical reasoning.

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