Question

If the statement $p \leftrightarrow (q \rightarrow p)$ is false, then true statement/statement pattern is

• A. $p$
• B. $p \rightarrow (p \vee \sim q)$
• C. $p \wedge (\sim p \wedge q)$
• D. $(p \vee \sim q) \rightarrow p$

Hint:

To solve problems involving logical statements, first determine the truth values of the basic propositions ($p, q$) that satisfy the given condition. Then, substitute these truth values into each option to find the one that results in a true statement. Remember the truth tables for conditional ($A \rightarrow B$ is false only if $A$ is true and $B$ is false) and biconditional ($A \leftrightarrow B$ is false if $A$ and $B$ have different truth values).

Answer 1

The Correct Option is B

Step 1: Determine the truth values of $p$ and $q$ for which $p \leftrightarrow (q \rightarrow p)$ is false. A biconditional statement $A \leftrightarrow B$ is false if and only if $A$ and $B$ have different truth values (one is true and the other is false). So, $p \leftrightarrow (q \rightarrow p)$ is false implies either: Case 1: $p$ is True and $(q \rightarrow p)$ is False. Case 2: $p$ is False and $(q \rightarrow p)$ is True. Let's analyze Case 1: $p=T$ and $(q \rightarrow p)=F$. For $(q \rightarrow p)$ to be False, $q$ must be True and $p$ must be False. This contradicts $p=T$. So Case 1 is not possible. Let's analyze Case 2: $p=F$ and $(q \rightarrow p)=T$. For $(q \rightarrow p)$ to be True, if $p=F$, then $q$ must be False. (If $q=T$, then $q \rightarrow p$ would be $T \rightarrow F = F$, which contradicts $(q \rightarrow p)=T$). Therefore, the only possibility is $p=F$ and $q=F$.
Step 2: Evaluate each option with $p=F$ and $q=F$. Option A: $p$ If $p=F$, then $p$ is false. Option B: $p \rightarrow (p \vee \sim q)$ Substitute $p=F$ and $q=F$: \[ F \rightarrow (F \vee \sim F) \] Since $\sim F = T$: \[ F \rightarrow (F \vee T) \] Since $F \vee T = T$: \[ F \rightarrow T \] This statement is True. Option C: $p \wedge (\sim p \wedge q)$ Substitute $p=F$ and $q=F$: \[ F \wedge (\sim F \wedge F) \] Since $\sim F = T$: \[ F \wedge (T \wedge F) \] Since $T \wedge F = F$: \[ F \wedge F \] This statement is False. Option D: $(p \vee \sim q) \rightarrow p$ Substitute $p=F$ and $q=F$: \[ (F \vee \sim F) \rightarrow F \] Since $\sim F = T$: \[ (F \vee T) \rightarrow F \] Since $F \vee T = T$: \[ T \rightarrow F \] This statement is False.
Step 3: Identify the true statement/statement pattern. From the evaluation, only Option B is true when $p=F$ and $q=F$.