Gate2018 cs Q61

61Consider 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?

Option : A
Explanation :
∀ are always True and ∃ are always False for empty sets.
So there exists at least one model with universe of size 3 (or less than). Therefore, option (A) is necessarily TRUE.

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