Question 1208962: Premise:
1.(∃x) (Ax • Bx) ∨ (∃x) (Cx ∨ Dx)
2.(∃x) (Ax ∨ Cx) ⊃ (x) Ex
3.~Em
Conclusion:
(∃x) Dx
Answer by textot(100)   (Show Source):
You can put this solution on YOUR website! **1. (∃x) (Ax • Bx) ∨ (∃x) (Cx ∨ Dx)**
Given
**2. (∃x) (Ax ∨ Cx) ⊃ (x) Ex**
Given
**3. ~Em**
Given
**4. ~(x) Ex**
Universal Instantiation (3)
**5. ~[(∃x) (Ax ∨ Cx)]**
Modus Tollens (2, 4)
**6. ~[(∃x) (Ax ∨ Cx)] ≡ [~(∃x) Ax ∧ ~(∃x) Cx]**
De Morgan's Law (Quantifier Form)
**7. ~(∃x) Ax ∧ ~(∃x) Cx**
Equivalence (5, 6)
**8. ~(∃x) Ax**
Simplification (7)
**9. ~[(∃x) (Ax • Bx)]**
Simplification (7)
**10. ~(∃x) Ax ∨ ~[(∃x) Bx]**
De Morgan's Law (Quantifier Form) (9)
**11. ~(∃x) Ax**
Simplification (10)
**12. (∃x) (Cx ∨ Dx)**
Disjunctive Syllogism (1, 11)
**13. (∃x) Cx ∨ (∃x) Dx**
Distributive Law (Quantifier Form) (12)
**14. ~(∃x) Cx**
Simplification (7)
**15. (∃x) Dx**
Disjunctive Syllogism (13, 14)
**Therefore, (∃x) Dx**
This derivation demonstrates that the conclusion (∃x) Dx can be derived from the given premises using the specified rules of inference and quantifier negation.
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