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Consider the first-order logic sentence 

F: ∀ x (∃ y R(x,y)). 

Q. Assuming non-empty logical domains, which of the sentences below are implied by F? 

I. ∃y (∃x R(x,y))
II. ∃y (∀x R(x,y))
III. ∀y (∃x R(x,y))
IV. ∼∃x (∀y R(x,y))
  • a)
    IV only
  • b)
    I and IV only
  • c)
    II only
  • d)
    II and III only
Correct answer is option 'B'. Can you explain this answer?
Verified Answer
Consider the first-order logic sentenceF: ∀ x (∃ y R(x,y))...
Given, first order logic sentence 
F: ∀x (∃y R(x, y)) with following sentences: 
(i) ∃y (∃x R(x, y)) is true. Because we have ∀x (∃y R(x, y)) → ∃x (∃y R(x, y)) → ∃y (∃x R(x, y)). 
(ii) ∃y (∀x R(x, y)) is false. Because we have ∀x (∃y R(x, y)) ← ∃y (∀x R(x, y)). 
(iii) ∀y (∃x R(x, y)) is false. Because for ∃y can not imply ∀y.
(iv) ∼∃x (∀y ∼R(x, y)) is true. Because ∼∃x (∀y ∼R(x, y)) = ∀x (∼ ∃y ∼R(x, y)) = ∀x (∃y ∼∼R(x, y)) = ∀x (∃y R(x, y)). 
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Consider the first-order logic sentenceF: ∀ x (∃ y R(x,y)).Q. Assuming non-empty logical domains, which of the sentences below are implied by F?I. ∃y (∃x R(x,y))II. ∃y (∀x R(x,y))III. ∀y (∃x R(x,y))IV. ∼∃x (∀y R(x,y))a)IV onlyb)I and IV onlyc)II onlyd)II and III onlyCorrect answer is option 'B'. Can you explain this answer?
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