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The negation of (p vv ~q) ^^ q is...

The negation of `(p vv ~q) ^^ q` is

A

`(~p vv q) ^^ ~q`

B

`(p ^^ ~q) vv q`

C

`(~p ^^ q) ^^ ~q`

D

`(p ^^ ~q) ^^ ~q`

Text Solution

AI Generated Solution

The correct Answer is:
To find the negation of the expression \( (p \lor \neg q) \land q \), we can follow these steps: ### Step 1: Identify the Original Expression The original expression we are negating is: \[ (p \lor \neg q) \land q \] ### Step 2: Apply De Morgan's Laws According to De Morgan's Laws, the negation of a conjunction is the disjunction of the negations. Therefore, we can express the negation as: \[ \neg((p \lor \neg q) \land q) = \neg(p \lor \neg q) \lor \neg q \] ### Step 3: Negate Each Component Now we need to negate each part of the expression: 1. Negate \( (p \lor \neg q) \): \[ \neg(p \lor \neg q) = \neg p \land q \] (Again, applying De Morgan's Laws) 2. Negate \( q \): \[ \neg q \] ### Step 4: Combine the Negated Components Now we combine the results from Step 3: \[ \neg((p \lor \neg q) \land q) = (\neg p \land q) \lor \neg q \] ### Step 5: Final Expression Thus, the negation of the original expression \( (p \lor \neg q) \land q \) is: \[ (\neg p \land q) \lor \neg q \] ### Conclusion The final answer is: \[ (\neg p \land q) \lor \neg q \]
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