Question

The statement \(p \rightarrow (q \rightarrow p)\) is equivalent to

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

Hint:

The statement \(p \rightarrow (q \rightarrow p)\) is a tautology, equivalent to \(p \rightarrow (p \vee q)\).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Use truth tables or logical equivalences. \(q \rightarrow p\) is equivalent to \(\neg q \vee p\). Then \(p \rightarrow (q \rightarrow p)\) is \(p \rightarrow (\neg q \vee p)\) which is \(\neg p \vee (\neg q \vee p) = (\neg p \vee p) \vee \neg q = T \vee \neg q = T\). So it's a tautology. Check options: (C) \(p \rightarrow (p \vee q) = \neg p \vee (p \vee q) = (\neg p \vee p) \vee q = T \vee q = T\). So it's also a tautology. Others are not always true.

Step 2: Detailed Explanation:
Truth table: For \(p\) true, \(q\) true: LHS: \(T \rightarrow (T \rightarrow T) = T \rightarrow T = T\). RHS (C): \(T \rightarrow (T \vee T) = T \rightarrow T = T\). All cases match. For \(p\) false, LHS: \(F \rightarrow (q \rightarrow F) = T\). RHS: \(F \rightarrow (F \vee q) = T\). So equivalent.

Step 3: Final Answer:
\(p \rightarrow (p \vee q)\), which corresponds to option (C).