GATE CS 2018 | Q 28 : Consider the first-order logic sentence 𝜑 ≡ ∃s∃t∃u∀v∀w∀x∀y 𝜓(s,t,u,v,w,x,y)

GATE CS 2018 | Question: 28
Consider the first-order logic sentence
𝜑≡∃s∃t∃u∀v∀w∀x∀y 𝜓(s,t,u,v,w,x,y)
where 𝜓(s,t,u,v,w,x,y) is a quantifier-free first-order logic formula using only predicate symbols, and possibly equality, but no function symbols. Suppose  𝜑  has a model with a universe containing 7 elements.
Which one of the following statements is necessarily true?
(A) There exists at least one model of  𝜑  with universe of size less than or equal to 3
(B) There exists no model of  𝜑  with universe of size less than or equal to 3
(C) There exists no model of  𝜑  with universe size of greater than 7
(D) Every model of  𝜑  has a universe of size equal to 7 

Ans: A
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0:40 Solution

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