Consider the given formulas below: F1: (p ⇔ q) Λ (~p ⇔ q) F2: (p V ~q) Λ (~p V…
Consider the given formulas below:
F1: (p ⇔ q) Λ (~p ⇔ q)
F2: (p V ~q) Λ (~p V q) Λ (~p V ~q)
Which of the following is/are correct:
Answer: C. F1 is Unsatisfiable and F2 is satisfiable — Analysis of F1: (p ⇔ q) ∧ (~p ⇔ q) Step 1: Consider the truth values of p and q. Case 1: p = true. For (p ⇔ q) to be true, q must be true. Then (~p ⇔ q)…
- A.
F1 and F2 both are satisfiable
- B.
F1 is valid and F2 is Satisfiable
- C.
F1 is Unsatisfiable and F2 is satisfiable
- D.
F1 and F2 both are unsatisfiable
Attempted by 30 students.
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Correct answer: C
Analysis of F1: (p ⇔ q) ∧ (~p ⇔ q)
Step 1: Consider the truth values of p and q.
Case 1: p = true. For (p ⇔ q) to be true, q must be true. Then (~p ⇔ q) becomes (false ⇔ true), which is false. So F1 is false.
Case 2: p = false. For (p ⇔ q) to be true, q must be false. Then (~p ⇔ q) becomes (true ⇔ false), which is false. So F1 is false.
Conclusion: F1 is unsatisfiable because no truth assignment makes it true.
Analysis of F2: (p ∨ ¬q) ∧ (¬p ∨ q) ∧ (¬p ∨ ¬q)
Step 1: Try p = false, q = false.
(p ∨ ¬q) = (false ∨ true) = true
(¬p ∨ q) = (true ∨ false) = true
(¬p ∨ ¬q) = (true ∨ true) = true
Conclusion: F2 is satisfiable because at least one truth assignment (p = false, q = false) makes it true.