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.

QUESTION A AND B ALREADY SOLVED. Solve part C, D, E using same method, and give some explantion for your answer. 

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.

• H(x): x is on the honor roll
• A(x): x received an A.
• D(x): x got a detention.

∀x (H(x) → A(x))
¬∃x (D(x) ∧ A(x))
∴ ¬∃x (D(x) ∧ H(x))

Valid.

1.	∀x (H(x) → A(x))	Hypothesis
2.	c is an arbitrary element	Element definition
3.	H(c) → A(c)	Universal instantiation, 1, 2
4.	¬∃x (D(x) ∧ A(x))	Hypothesis
5.	∀x ¬(D(x) ∧ A(x))	De Morgan's law, 4
6.	¬(D(c) ∧ A(c))	Universal instantiation, 2, 5
7.	¬D(c) ∨ ¬A(c)	De Morgan's law, 6
8.	¬A(c) ∨ ¬D(c)	Commutative law, 7
9.	¬H(c) ∨ A(c)	Conditional identity, 3
10.	A(c) ∨ ¬H(c)	Commutative law, 9
11.	¬D(c) ∨ ¬H(c)	Resolution, 8, 10
12.	¬(D(c) ∧ H(c))	De Morgan's law, 11
13.	∀x ¬(D(x) ∧ H(x))	Universal generalization, 2, 12
14.	¬∃x (D(x) ∧ H(x))	De Morgan's law, 13

(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.

• M(x): x missed class
• A(x): x received an A.
• D(x): x got a detention.

¬∃x (A(x) ∧ M(x))
¬∃x (D(x) ∧ A(x))
∴ ¬∃x (D(x) ∧ M(x))

The argument is not valid. Consider a class that consists of a single student named Frank. If M(Frank) = D(Frank) = T and A(Frank) = F, then the hypotheses are all true and the conclusion is false. In other words, Frank got a detention, missed class, and did not get an A.

(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.