GATE | GATE CS 2018 | Question 54

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

Answer

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