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

Answer: (C)

Both P1 and P2 are FALSE

Answer 2

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.