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p ∧ q is logically equivalent to ________
- a)¬ (p → ¬q)
- b)(p → ¬q)
- c)(¬p → ¬q)
- d)(¬p → q)
Correct answer is option 'A'. Can you explain this answer?
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p ∧ q is logically equivalent to ________a)¬ (p → ¬q)...
Understanding Logical Equivalence
In propositional logic, two statements are logically equivalent if they have the same truth value in every possible scenario. Here, we will explore why "p ∧ q" is logically equivalent to "¬(p → ¬q".
Analyzing the Options
- Option A: ¬(p → ¬q)
- This expression can be rewritten using logical implications. The expression "p → ¬q" is equivalent to "¬p ∨ ¬q". Therefore, "¬(p → ¬q)" can be rephrased as "¬(¬p ∨ ¬q)".
- Applying De Morgan's laws, we can simplify this to "p ∧ q". Thus, "¬(p → ¬q)" is logically equivalent to "p ∧ q", confirming that Option A is correct.
- Option B: (p → ¬q)
- This suggests that if p is true, then q must be false, which does not reflect the conjunction of p and q.
- Option C: (¬p → ¬q)
- This means if p is false, then q must also be false, which is not equivalent to p and q being true simultaneously.
- Option D: (¬p → q)
- This indicates that if p is false, then q is true, which again does not capture the essence of "p ∧ q".
Conclusion
Based on the analysis above, the correct answer is indeed option A: "¬(p → ¬q)", as it is logically equivalent to "p ∧ q" using fundamental logical transformations and laws. This understanding is crucial for tackling problems in Computer Science Engineering and logic-based reasoning.
In propositional logic, two statements are logically equivalent if they have the same truth value in every possible scenario. Here, we will explore why "p ∧ q" is logically equivalent to "¬(p → ¬q".
Analyzing the Options
- Option A: ¬(p → ¬q)
- This expression can be rewritten using logical implications. The expression "p → ¬q" is equivalent to "¬p ∨ ¬q". Therefore, "¬(p → ¬q)" can be rephrased as "¬(¬p ∨ ¬q)".
- Applying De Morgan's laws, we can simplify this to "p ∧ q". Thus, "¬(p → ¬q)" is logically equivalent to "p ∧ q", confirming that Option A is correct.
- Option B: (p → ¬q)
- This suggests that if p is true, then q must be false, which does not reflect the conjunction of p and q.
- Option C: (¬p → ¬q)
- This means if p is false, then q must also be false, which is not equivalent to p and q being true simultaneously.
- Option D: (¬p → q)
- This indicates that if p is false, then q is true, which again does not capture the essence of "p ∧ q".
Conclusion
Based on the analysis above, the correct answer is indeed option A: "¬(p → ¬q)", as it is logically equivalent to "p ∧ q" using fundamental logical transformations and laws. This understanding is crucial for tackling problems in Computer Science Engineering and logic-based reasoning.
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Community Answer
p ∧ q is logically equivalent to ________a)¬ (p → ¬q)...
(p ∧ q) ↔ (¬(p → ¬q)) is tautology.
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