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Which one of the following is NOT logically equivalent to ¬ ∃x (∀ y (α) ∧ ∀ z(β))?

Which one of the following is NOT logically equivalent to ¬ ∃x (∀ y (α) ∧ ∀ z(β))? Correct Answer ∀x (∃y (¬ α) → ∃z (¬ β))

The correct answer is "option 1 and 4".

EXPLANATION:

Option 1:

 ∀x (∃z (¬ β) → ∀y (α))

→ ∀x (¬∃z (¬ β) v ∀y (α))

→ ∀x (∀z (β) v y (α))

→ ¬∃x ¬ (∀z (β) v y (α))

→ ¬∃x (¬∀z (β) ∧ ¬y (α))

Hence, ∀x (∃z (¬ β) → ∀y (α)) is not equivalent to ¬ ∃x (∀ y (α) ∧ ∀ z(β)).

Option 2:

∀x (∀z (β) → ∃y (¬ α))

→ ∀x (¬∀z (β) v ∃y (¬ α))

→ ¬∃x¬ (¬∀z (β) v ∃y (¬ α))

→ ¬∃x (∀z (β) v ∀y (α))

Hence, ∀x (∀z (β) → ∃y (¬ α)) is equivalent to ¬ ∃x (∀ y (α) ∧ ∀ z(β)).

Option 3:

∀x (∀y (α) → ∃z (¬ β))

→ ∀x (¬∀y (α) v ∃z (¬ β))

→ ¬∃x¬ (¬∀y (α) v ∃z (¬ β))

→ ¬∃x (∀y (α) ^ ∀z (β))

Hence, ∀x (∀y (α) → ∃z (¬ β)) is equivalent to ¬ ∃x (∀ y (α) ∧ ∀ z(β)).

Option 4:

∀x (∃y (¬ α) → ∃z (¬ β))

→ ∀x (¬∃y (¬ α) v ∃z (¬ β))

→ ∀x (∀y (α) v ∃z (¬ β))

→ ¬∃x¬ (∀y (α) v ∃z (¬ β))

→ ¬∃x (¬∀y (α) ∧ ¬∃z (¬ β))

→ ¬∃x (¬∀y (α) ∧ ∀z (β))

Hence, ∀x (∃y (¬ α) → ∃z (¬ β)) is not equivalent to ¬ ∃x (∀ y (α) ∧ ∀ z(β)).

Hence, the correct answer is "option 1 and 4".

NOTE:
Marks are given to all for this question in the official GATE CS 2013

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