Question

Match LIST-I with LIST-II \[\begin{array}{|c|c|}\hline \text{LIST-I (Standard logical equivalence)} & \text{LIST-II (Theorem)} \\ \hline \text{A. } (\alpha \iff \beta) \equiv (\beta \iff \neg \alpha) & \text{IV. Contraposition} \\ \hline \text{B. } (\alpha \iff \beta) \equiv ((\alpha \iff \beta) \land (\beta \iff \alpha)) & \text{I. Biconditional elimination} \\ \hline \text{C. } (\alpha \lor \beta) \equiv (\alpha \land \neg \beta) & \text{III. Distributivity of } \lor \text{ over } \land \\ \hline \text{D. } (\alpha \lor \beta) \equiv (\neg \alpha \lor \neg \beta) & \text{II. De Morgan's law} \\ \hline \end{array}\] Choose the correct answer from the options given below:

• 1. A - I, B - II, C - III, D - IV
• 2. A - I, B - II, C - IV, D - III
• 3. A - IV, B - II, C - I, D - III
• 4. A - III, B - IV, C - I, D - II

Hint:

Understanding logical equivalences is crucial in simplifying logical expressions and proofs.

Answer 1

The Correct Option is A

- **A** corresponds to **I** because **Biconditional elimination** refers to eliminating a biconditional expression and converting it into two conditional expressions. The equivalence is \((\alpha \iff \beta) \equiv (\beta \iff \neg \alpha)\). - **B** corresponds to **II** because **De Morgan's law** deals with negating and distributing logical operations such as conjunction and disjunction. - **C** corresponds to **III** because **Distributivity of \(\lor\) over \(\land\)** states that a logical or (\(\lor\)) distributes over a logical and (\(\land\)), as shown in the equivalence. - **D** corresponds to **IV** because **Contraposition** involves flipping and negating both parts of an implication. Step 2: Conclusion. The correct match is **A - I, B - II, C - III, D - IV**.