Computer Science Engineering (CSE) Exam > Computer Science Engineering (CSE) Questions > Let P and Q be two propositions, ¬(P&harr...
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Let P and Q be two propositions, ¬(P↔Q) is equivalent to:
- P ↔ ¬ Q
- ¬ P↔ Q
- ¬ P ↔ ¬ Q
- Q ↔ P
- a)1 and 2
- b)2 and 3
- c)3 and 4
- d)None of the above
Correct answer is option 'A'. Can you explain this answer?
Most Upvoted Answer
Let P and Q be two propositions, ¬(P↔Q) is equivalent to: P &...
We know that
Both statements (1) and (2) are correct.
So, option (1) is correct.
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Community Answer
Let P and Q be two propositions, ¬(P↔Q) is equivalent to: P &...
Understanding the Problem
The proposition ¬(P↔Q) means "not (P if and only if Q)." We need to determine its equivalence with the provided options.
Logical Equivalence Explanation
1. Definition of Biconditional (P ↔ Q):
- P ↔ Q is true if both P and Q are either true or false. It is false if one is true and the other is false.
2. Negation of Biconditional:
- ¬(P ↔ Q) is true when P and Q have different truth values (i.e., one is true, and the other is false).
3. Analyzing the Options:
- Option 1: P ↔ ¬Q
- This means P is true if and only if Q is false. This is equivalent to saying that P and Q have different truth values.
- Option 2: ¬ P ↔ Q
- This means "not P" is true if and only if Q is true, which also indicates that P and Q have different truth values.
- Option 3: ¬ P ↔ ¬ Q
- This means "not P" is true if and only if "not Q" is true, indicating that both are either true or false. Therefore, they do not have different truth values.
- Option 4: Q ↔ P
- This is equivalent to P ↔ Q, and thus is not negated.
Conclusion
From the analysis, we see that ¬(P↔Q) is equivalent to:
- Option 1 (P ↔ ¬Q)
- Option 2 (¬ P ↔ Q)
Hence, the correct answer is option 'A' (1 and 2).
This demonstrates that the negation of a biconditional statement effectively captures the conditions where the propositions differ in truth value.
The proposition ¬(P↔Q) means "not (P if and only if Q)." We need to determine its equivalence with the provided options.
Logical Equivalence Explanation
1. Definition of Biconditional (P ↔ Q):
- P ↔ Q is true if both P and Q are either true or false. It is false if one is true and the other is false.
2. Negation of Biconditional:
- ¬(P ↔ Q) is true when P and Q have different truth values (i.e., one is true, and the other is false).
3. Analyzing the Options:
- Option 1: P ↔ ¬Q
- This means P is true if and only if Q is false. This is equivalent to saying that P and Q have different truth values.
- Option 2: ¬ P ↔ Q
- This means "not P" is true if and only if Q is true, which also indicates that P and Q have different truth values.
- Option 3: ¬ P ↔ ¬ Q
- This means "not P" is true if and only if "not Q" is true, indicating that both are either true or false. Therefore, they do not have different truth values.
- Option 4: Q ↔ P
- This is equivalent to P ↔ Q, and thus is not negated.
Conclusion
From the analysis, we see that ¬(P↔Q) is equivalent to:
- Option 1 (P ↔ ¬Q)
- Option 2 (¬ P ↔ Q)
Hence, the correct answer is option 'A' (1 and 2).
This demonstrates that the negation of a biconditional statement effectively captures the conditions where the propositions differ in truth value.
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