Question

Determine and prove whether an argument in English is valid or invalid. About 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.

Answer 1

Answer:

Each understudy on the respect roll got an A.  

No understudy who got a confinement got an A.  

No understudy who got a confinement is on the respect roll.  

No understudy who got an A missed class.  

No understudy who got a confinement got an A.  

No understudy who got a confinement missed class  

Explanation:

M(x): x missed class  

An (x): x got an A.  

D(x): x got a confinement.  

¬∃x (A(x) ∧ M(x))  

¬∃x (D(x) ∧ A(x))  

∴ ¬∃x (D(x) ∧ M(x))  

The conflict isn't considerable. Consider a class that includes a lone understudy named Frank. If M(Frank) = D(Frank) = T and A(Frank) = F, by then the hypotheses are overall evident and the end is counterfeit. Toward the day's end, Frank got a control, missed class, and didn't get an A.  

Each understudy who missed class got a confinement.  

Penelope is an understudy in the class.  

Penelope got a confinement.  

Penelope missed class.  

M(x): x missed class  

S(x): x is an understudy in the class.  

D(x): x got a confinement.  

Each understudy who missed class got a confinement.  

Penelope is an understudy in the class.  

Penelope didn't miss class.  

Penelope didn't get imprisonment.  

M(x): x missed class  

S(x): x is an understudy in the class.  

D(x): x got imprisonment.  

Each understudy who missed class or got imprisonment didn't get an A.  

Penelope is an understudy in the class.  

Penelope got an A.  

Penelope didn't get repression.  

M(x): x missed class  

S(x): x is an understudy in the class.  

D(x): x got a repression.  

An (ax): x got an A.  

H(x): x is on the regard roll  

An (x): x got an A.  

D(x): x got a repression.  

∀x (H(x) → A(x)) a  

¬∃x (D(x) ∧ A(x))  

∴ ¬∃x (D(x) ∧ H(x))  

Real.  

1. ∀x (H(x) → A(x)) Hypothesis  

2. c is a self-self-assured element Element definition  

3. H(c) → A(c) Universal dispatch, 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 dispatch, 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 character, 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 hypothesis, 2, 12  

14. ¬∃x (D(x) ∧ H(x)) De Morgan's law, 13