Computer Information Systems

To derive ∀ 𝑥 ( ¬ 𝑃 ( 𝑥 ) ∨ 𝑄 ( 𝑥 ) ) ∀ x ( ¬ P ( x ) ∨ Q ( x )) from ¬ ∃ 𝑥 ( 𝑃 ( 𝑥 ) ∧ ¬ 𝑄 ( 𝑥 ) ) ¬∃ x ( P ( x ) ∧ ¬ Q ( x )) , we'll use a formal proof in natural deduction. ¬ ∃ 𝑥 ( 𝑃 ( 𝑥 ) ∧ ¬ 𝑄 ( 𝑥 ) ) ¬∃ x ( P ( x ) ∧ ¬ Q ( x )) Premise | Assume ∃ 𝑥 ( 𝑃 ( 𝑥 ) ∧ ¬ 𝑄 ( 𝑥 ) ) ∃ x ( P ( x ) ∧ ¬ Q ( x )) Assumption for Negation Introduction | Let 𝑎 a be such that 𝑃 ( 𝑎 ) ∧ ¬ 𝑄 ( 𝑎 ) P ( a ) ∧ ¬ Q ( a ) Existential Instantiation | 𝑃 ( 𝑎 ) P ( a ) Conjunction Elimination (from 3) | ¬ 𝑄 ( 𝑎 ) ¬ Q ( a ) Conjunction Elimination (from 3) | 𝑃 ( 𝑎 ) ∧ ¬ 𝑄 ( 𝑎 ) P ( a ) ∧ ¬ Q ( a ) Conjunction Introduction (4, 5) | ∃ 𝑥 ( 𝑃 ( 𝑥 ) ∧ ¬ 𝑄 ( 𝑥 ) ) ∃ x ( P ( x ) ∧ ¬ Q ( x )) Existential Introduction (6) | Contradiction (2, 7) Reiteration (2, 7) ¬ ∃ 𝑥 ( 𝑃 ( 𝑥 ) ∧ ¬ 𝑄 ( 𝑥 ) ) ¬∃ x ( P ( x ) ∧ ¬ Q ( x )) Negation Introduction (2-8) Assume 𝑥 x is arbitrary Universal Generalization Preparation | Assume 𝑃 ( 𝑥 ) P ( x ) Assumption for Disjunction Introduction | ¬ 𝑃 ( 𝑥 ) ∨ 𝑄 ( 𝑥 ) ¬ P ( x ) ∨ Q ( x ) Disjunction Introduction (from 11) Assume ¬ 𝑃 ( 𝑥 ) ¬ P ( x ) Assumption for Disjunction Introduction ¬ 𝑃 ( 𝑥 ) ∨ 𝑄 ( 𝑥 ) ¬ P ( x ) ∨ Q ( x ) Disjunction Introduction (from 13) ¬ 𝑃 ( 𝑥 ) ∨ 𝑄 ( 𝑥 ) ¬ P ( x ) ∨ Q ( x ) Cases (12, 14) ∀ 𝑥 ( ¬ 𝑃 ( 𝑥 ) ∨ 𝑄 ( 𝑥 ) ) ∀ x ( ¬ P ( x ) ∨ Q ( x )) Universal Generalization (10-15) The proof demonstrates that ∀ 𝑥 ( ¬ 𝑃 ( 𝑥 ) ∨ 𝑄 ( 𝑥 ) ) ∀ x ( ¬ P ( x ) ∨ Q ( x )) follows logically from ¬ ∃ 𝑥 ( 𝑃 ( 𝑥 ) ∧ ¬ 𝑄 ( 𝑥 ) ) ¬∃ x ( P ( x ) ∧ ¬ Q ( x )) . The essential idea is that if it's not the case that there exists an 𝑥 x such that 𝑃 ( 𝑥 ) ∧ ¬ 𝑄 ( 𝑥 ) P ( x ) ∧ ¬ Q ( x ) , then for every 𝑥 x , it must be the case that either ¬ 𝑃 ( 𝑥 ) ¬ P ( x ) or 𝑄 ( 𝑥 ) Q ( x )

Tomorrow Happy Karwa Chauth from Isolated Study! Study smarter this festive season with IsolatedAI.Try AI free →

Computer science

Isolated Study

Answer

Student’s rating for this solution:

★ ★ ★ ★ ★ (5/5)

YOU HAVE TO REGISTER FOR ANSWER

AI tools

Study help

Free tools

Earn & more

Computer science

Computer science

Anu Bhardwaj

Answer

Extracted Transcript from Images

AI

Study help

Earn & more

Company