1. The problem statement, all variables and given/known data In the questions below suppose the variable x represents students and y represents courses, and: U(y): y is an upper-level course M(y): y is a math course F(x): x is a freshman A(x): x is a part-time student T(x,y): student x is taking course y. Write the statement using these predicates and any needed quantifiers. 1. There is a part-time student who is not taking any math course. Answer: ∃x∀y[A(x) ∧ (M(y) → ¬T(x,y))]. 2. Every part-time freshman is taking some upper-level course. Answer: ∀x∃y[(F(x) ∧ A(x)) → (U(y) ∧ T(x,y))]. 2. Relevant equations None 3. The attempt at a solution I'm having trouble rewriting statements using predicates and quantifiers. I can't seem to find much information on restricted quantifiers especially with multiple quantifiers. It seems like the whole purpose of restricted quantifiers is to minimize the domain when rewriting the statements, but I can't seem to translate it well. I believe I understand the first one: There exists a student x, that for all courses y, if x is a part time student and y is a math course, then x is not taking course y. So the "If" portion minimizes the domain. The second one is where I get lost. I thought the answer would be: ∀x∃y[(F(x) ∧ A(x)) ∧ U(y) → T(x,y)] For all students x, there exists a course y such that if x is a freshmen and x is a part-time student and y is a upper-level course, then x is taking y. I don't understand why this is not correct. The "if" restricts the domain to part-time, freshmen and upper level courses.