Determine and prove whether an argument in English is valid or invalid. 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. PLEASE solve part C, D, E using same method, and give some explantion for your answer. THANKS! 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.

Hello! I'm happy to help you out in this matter. Following is the answer to your argument :

(c) Every student who received an A is on the honor roll.
H(x): x is on the honor roll
A(x): x received an A.
This argument is invalid. We can show this by giving a counterexample. For example, consider a scenario where there is a student named John who received an A, but he is not on the honor roll. This scenario contradicts the conclusion of the argument, which states that every student who received an A is on the honor roll. Therefore, the argument is invalid.

(d) No student who is on the honor roll received a detention.
H(x): x is on the honor roll
D(x): x got a detention.
This argument is valid. We can prove its validity using the rules of inference. The argument can be written symbolically as ¬∃x(H(x) ∧ D(x)), which can be rewritten using De Morgan's law as ∀x(¬H(x) ∨ ¬D(x)). This is a valid argument form known as the rule of inference called universal instantiation.

(e) Every student who did not receive an A got a detention.
A(x): x received an A.
D(x): x got a detention.
This argument is invalid. We can show this by giving a counterexample. For example, consider a scenario where there is a student named Sarah who did not receive an A, but she did not get a detention either. This scenario contradicts the conclusion of the argument, which states that every student who did not receive an A got a detention. Therefore, the argument is invalid.

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