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COMPAILER-DESIGN-10CS63-VTU-NOTES-UNIT-2-->View question

## 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 of size greater than 7.(D) Every model of φ has a universe of size equal to 7.

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Quick logic review -

$\alpha :\mathrm{\forall }x\mathrm{\exists }y\phantom{\rule{.5em}{0ex}}y

Is $\alpha$ true for domain of all integers ?, Yes it is true. You pick any number $x$, I can always give you $y$ that is less than your number $x$

Is $\alpha$ true for domain of  Non Negative integers $\left\{0,1,2,3,\dots \right\}$ ? No,  it is not true. (You pick any number $x$) If you pick $0$ then I can not give you $y$ which is less than $0$.

Definition of  - Domain for which my sentence is true. For above sentence $\alpha$all integers is model and there can be many other models, like -  real numbers.

(Definition of Domain for which my sentence is False.)

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