Prove whether each argument is valid or invalid. First find the form of the argument by defining predicates and expressing the hypotheses and the conclusion using the predicates. If the argument is valid, then use the rules of inference to prove that the form is valid. If the argument is invalid, give values for the predicates you defined for a small domain that demonstrate the argument is invalid. The domain for each problem is the set of students in a class.

(a) Every student on the honor roll received an A.

No student who got a detention received an A.

No student who got a detention is on the honor roll.

(b) No student who got an A missed class.

No student who got a detention received an A.

No student who got a detention missed class.

(c) Every student who missed class got a detention.

Penelope is a student in the class.

Penelope got a detention.

Penelope missed class.

(d) Every student who missed class got a detention

Penelope is a student in the class.

Penelope did not miss class.

Penelope did not get a detention.

(e) Every student who missed class or got a detention did not get an A.

Penelope is a student in the class.

Penelope got an A.

Penelope did not get a detention.

Gravel is being dumped from a conveyor belt at a rate of 10 ft/min. It forms a pile in the shape of a right circular cone whose base diameter and height are always the same. How fast is the height of the pile increasing when the pile is 18 ft high? The height is increasing at 180 ft/min.