If the statement \( p \leftrightarrow (q \rightarrow p) \) is false, then the true statement is:
Show Hint
When analyzing logical biconditional statements, test various truth values for the components involved to identify when the statement becomes false. Then compare this to the given options.
Let's analyze the given statement \( p \leftrightarrow (q \rightarrow p) \). This statement is a biconditional statement which is false only when one side is true and the other is false.
First, consider \( p \) as true and \( q \) as false. Then \( (q \rightarrow p) \) will be true, so \( p \leftrightarrow (q \rightarrow p) \) becomes true \( \leftrightarrow \) true, which is true. Thus, \( p \) true and \( q \) false doesn't make the given statement false.
Next, consider \( p \) as false and \( q \) as true. Then \( (q \rightarrow p) \) will be false, so \( p \leftrightarrow (q \rightarrow p) \) becomes false \( \leftrightarrow \) false, which is true. Therefore, \( p \) false and \( q \) true doesn't make the given statement false either.
Now, let's consider \( p \) as false and \( q \) as false. Then \( (q \rightarrow p) \) will be true. Therefore, \( p \leftrightarrow (q \rightarrow p) \) becomes false \( \leftrightarrow \) true, which is false. This satisfies the condition of the question.
Now, let's check the options for \( p \) as false and \( q \) as false:
A. \( p \) is false.
B. \( p \rightarrow (p \vee \sim q) \) becomes false \( \rightarrow \) (false \( \vee \) true), which is false \( \rightarrow \) true = true.
C. \( p \wedge (\sim p q) \) becomes false \( \wedge \) (true \( \wedge \) false) = false.
D. \( (p \vee \sim q) \rightarrow p \) becomes (false \( \vee \) true) \( \rightarrow \) false = true \( \rightarrow \) false = false.
Option B is the only statement which is true when \( p \) is false and \( q \) is false.
