Question

Consider the following two propositions:
\(P_1 : ~ (p → ~ q)\)
\(P_2: (p ∧ ~q) ∧ ((-~p) ∨ q)\)
If the proposition \(p → ((~p) ∨ q)\) is evaluated as FALSE, then:

• A. P₁ is TRUE and P₂ is FALSE
• B. P₁ is FALSE and P₂ is TRUE
• C. Both P₁ and P₂ are FALSE
• D. Both P₁ and P₂ are TRUE

Answer 1

The Correct Option is C

Given that: \(p → (~ p∨q)\) is false.
\(⇒ ~ p ∨ q\) is false and \(p\) is true.
Now \(p \)is True.
\(~ T∨q = F\)
\(F∨q = F\)
\(⇒\) \(q\) is false
\(P_1: ~ (T → ~ F) ≡ ~ (T → T) ≡ False.\)
\(P_2: (T∧~ F) ∧(~ T ∨F) ≡ (T∧T) ∧(F∨F)\)
\(≡ T∧ F ≡\) False
So, the correct option is (C): Both \(P_1\) and \(P_2\) are FALSE.