Question

Consider the following two propositions: \[ P_1: \neg (p \rightarrow \neg q) \] \[ P_2: (p \wedge \neg q) \wedge ((\neg p) \vee q) \] If the proposition \( p \rightarrow ((\neg p) \vee q) \) is evaluated as FALSE, then:

• A. \( P_1 \) is TRUE and \( P_2 \) is FALSE
• B. \( P_1 \) is FALSE and \( P_2 \) is TRUE
• C. Both \( P_1 \) and \( P_2 \) are FALSE
• D. Both \( P_1 \) and \( P_2 \) are TRUE

Hint:

Constructing a truth table simplifies logical evaluation.

Answer 1

The Correct Option is C

We begin by constructing a truth table for the given expressions. The statement \( p \rightarrow ((\neg p) \vee q) \) is FALSE only when \( p = T \) and \( q = F \), which gives:

\[ p \rightarrow ((\neg p) \vee q) = F \] This condition leads to \( P_1 \) and \( P_2 \) both being FALSE.