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Justify (argue, prove, ...) that the following formula is valid Ha A(x) V Va В(г) — Vx (A(х) V B(х))

Justify (argue, prove, ...) that the following formula is valid Ha A(x) V Va В(г) — Vx (A(х) V B(х))

Database System Concepts
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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**Statement to Justify and Prove:** The logical formula to be proven valid is: ∃x A(x) ∨ ∀x B(x) → ∀x (A(x) ∨ B(x)) This formula involves a combination of existential quantifiers (`∃`), universal quantifiers (`∀`), and logical operators (disjunction `∨` and implication `→`). The goal is to show how the separate quantified predicates A(x) and B(x) together imply the universally quantified disjunction of these predicates. ### Explanation of Components: 1. **∃x A(x)**: There exists some element in the domain such that the predicate A(x) is true. 2. **∀x B(x)**: For all elements in the domain, the predicate B(x) is true. 3. **∨**: Logical OR, asserting that at least one of the statements is true. 4. **→**: Logical implication, meaning if the left side is true, then the right side must also be true. 5. **∀x (A(x) ∨ B(x))**: For all elements in the domain, either A(x) or B(x) (or both) is true. ### Steps for Justification: - **Assume** the left side: Assume that either there exists an x such that A(x) is true, or for all x, B(x) is true. - **Show** the right side: Demonstrate that for all x, A(x) or B(x) must be true. This task can involve a proof through logical reasoning, using known laws such as distributive laws or other rules within predicate logic, possibly using a formal proof system like natural deduction or truth tables for illustration.
Transcribed Image Text:**Statement to Justify and Prove:** The logical formula to be proven valid is: ∃x A(x) ∨ ∀x B(x) → ∀x (A(x) ∨ B(x)) This formula involves a combination of existential quantifiers (`∃`), universal quantifiers (`∀`), and logical operators (disjunction `∨` and implication `→`). The goal is to show how the separate quantified predicates A(x) and B(x) together imply the universally quantified disjunction of these predicates. ### Explanation of Components: 1. **∃x A(x)**: There exists some element in the domain such that the predicate A(x) is true. 2. **∀x B(x)**: For all elements in the domain, the predicate B(x) is true. 3. **∨**: Logical OR, asserting that at least one of the statements is true. 4. **→**: Logical implication, meaning if the left side is true, then the right side must also be true. 5. **∀x (A(x) ∨ B(x))**: For all elements in the domain, either A(x) or B(x) (or both) is true. ### Steps for Justification: - **Assume** the left side: Assume that either there exists an x such that A(x) is true, or for all x, B(x) is true. - **Show** the right side: Demonstrate that for all x, A(x) or B(x) must be true. This task can involve a proof through logical reasoning, using known laws such as distributive laws or other rules within predicate logic, possibly using a formal proof system like natural deduction or truth tables for illustration.
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