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. Hypotheses: a. Every student who missed class or got a detention did not get an A. b. Penelope is a student in the class. c. Penelope got an A. Conclusion: Penelope did not get a detention
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.
Hypotheses:
a. Every student who missed class or got a detention did not get an A.
b. Penelope is a student in the class.
c. Penelope got an A.
Conclusion:
Penelope did not get a detention
Let us define the predicates
S(x) : x is a student in the class
M(x) : x missed class
D(x) : x gets a detention
A(x) : x gets an A
Then,
Premise-a (P1)
∀x [ S(x) ∧ (M(x) ∨ D(x)) => ~A(x) ]
Premise-b (P2)
S(Penelope)
Premise-c (P3)
A(Penelope)
Conclusion (G)
~D(Penelope)
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