What is the first order predicate calculus statement equivalent to the…

Correct formalization: ∀x (teacher(x) → ∃y (student(y) ∧ likes(y,x)))

• Meaning of each part:

  • The outer universal quantifier ∀x applies the statement to every individual x.

  • The implication teacher(x) → ... ensures we only require the rest when x is a teacher.

  • The existential ∃y (student(y) ∧ likes(y,x)) requires there is some y who is a student and who likes x.

• Common pitfalls and why they are wrong:

  • Using ∃y (student(y) → likes(y,x)) is incorrect because an implication can be true for a non-student y, so it fails to guarantee the chosen y is a student who likes x.

  • Swapping quantifiers to ∃y ∀x ... changes the meaning to 'there exists one student who likes every teacher', which is stronger than the intended statement.

  • Putting a conjunction directly under ∀x as in ∀x (teacher(x) ∧ ...) incorrectly asserts that every object in the domain is a teacher.