This questin is asked in GATE EXAM.
Consider the first-order logic sentence φ≡∃s∃t∃u∀v∀w∀x∀yψ(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
My take- For empty set universal quantifiers are always true while existential quantifier are always false, hence, there exist at least one model with 3 elements, as it is given equality is also possible hence model is also possible for less than three elements, thus OPTION A.
Am I correct?
Also, from where can I study such concepts!