Logical Equivalents

In the study of logic, determining whether pairs of statements are logically equivalent is vital for understanding logical implications and relationships. Let's analyze the pairs presented:

Statement Analysis

1. Pair A: ( p \land q ) and ( p \vee q )

   • Interpretation:
     • ( p \land q ): Both ( p ) and ( q ) must be true.
     • ( p \vee q ): Either ( p ) is true, ( q ) is true, or both are true.
   • Conclusion: These statements are not equivalent.

2. Pair B: ( p \rightarrow q ) and ( \sim p \vee q )

   • Interpretation:
     • ( p \rightarrow q ): If ( p ) is true, then ( q ) is also true. If ( p ) is false, the statement holds regardless of ( q )'s truth value.
     • ( \sim p \vee q ): Either ( p ) is false or ( q ) is true.
   • Conclusion: According to the rules of logical equivalence, ( p \rightarrow q ) is indeed equivalent to ( \sim p \vee q ).

3. Pair C: ( \sim(p \vee q) ) and ( \sim p \land \sim q )

   • Interpretation:
     • ( \sim(p \vee q) ): Neither ( p ) nor ( q ) is true.
     • ( \sim p \land \sim q ): Both ( p ) is false and ( q ) is false.
   • Conclusion: By De Morgan's Laws, this pair is equivalent. Specifically, ( \sim(p \vee q) ) is equivalent to ( \sim p \land \sim q ).

Overview of Logical Equivalence

Based on our analyses:

• Pair A: Not Equivalent
• Pair B: Equivalent ( (p \rightarrow q \text{ is equivalent to } \sim p \vee q) )
• Pair C: Equivalent ( (\sim(p \vee q) \text{ is equivalent to } \sim p \land \sim q) )

Final Conclusion

Given the results from our evaluations:

• Option D is correct: Both B and C represent logically equivalent pairs. Thus, the answer is D.

Understanding and identifying these logical equivalences is crucial in both formal proofs and in practical applications such as computer science, mathematics, and philosophy.