Question

The correct logical equivalence from the following is /are ___
(I) \(p\rightarrow (q\rightarrow r)\equiv (p∧q)\rightarrow r\)
(II) \((p\rightarrow q)\rightarrow r\equiv p\rightarrow (q∨r)\)
(III) \((p\rightarrow q)\rightarrow r\equiv (p\rightarrow r)∧(\sim q\rightarrow r)\)
(IV) \(p\rightarrow (q\rightarrow r)\equiv q\rightarrow (p\rightarrow r)\)

• A. only (I) and (II)
• B. only (III) and (IV)
• C. only (II) and (IV)
• D. only (I) and (IV)

Hint:

Convert each implication into an or statement and compare.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
Use \(p \to q \equiv \sim p \vee q\) to rewrite each statement and compare.

Step 2: Key Formula or Approach:
\(p \to (q \to r) \equiv \sim p \vee \sim q \vee r\) and \((p \to q) \to r \equiv (p \wedge \sim q) \vee r\).

Step 3: Detailed Explanation:
Statement (I): \(p \to (q \to r) \equiv \sim p \vee (\sim q \vee r)\), and \((p \wedge q) \to r \equiv \sim p \vee \sim q \vee r\). These are the same, so (I) is TRUE.
Statement (II): \((p \to q) \to r \equiv (p \wedge \sim q) \vee r\), but \(p \to (q \vee r) \equiv \sim p \vee q \vee r\). They differ (take \(p\) false, \(r\) false: left is false, right is true), so (II) is FALSE.
Statement (III): \((p \to r) \wedge (\sim q \to r) \equiv (\sim p \vee r) \wedge (q \vee r) \equiv (\sim p \wedge q) \vee r\), which differs from \((p \wedge \sim q) \vee r\). So (III) is FALSE.
Statement (IV): both sides are \(\sim p \vee \sim q \vee r\), so (IV) is TRUE.

Final Answer:
Only (I) and (IV) are correct, option (D). \[ \boxed{\text{only (I) and (IV)}} \]