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

A truth table is constructed initially for the provided expressions. The statement \( p \rightarrow ((eg p) \vee q) \) evaluates to FALSE exclusively when \( p = T \) and \( q = F \). This is represented as:

\[ p \rightarrow ((eg p) \vee q) = F \] Consequently, both \( P_1 \) and \( P_2 \) are FALSE under this circumstance.