Negation of \( p \land (q \land \neg (p \land q)) \) is:}
Negation of \( p \land (q \land \neg (p \land q)) \) is:}
- ~(p∨q)
- p∨q
- (~(p∧q))∧q
- (~(p∧q))∨p
The Correct Option is D
Solution and Explanation
To find the negation of the statement (¬p∧q)∨(p∧¬q), we will follow the steps of logical negation and apply De Morgan's laws.
1. Write down the original statement:
S=(¬p∧q)∨(p∧¬q)
2. Apply negation to the entire statement:
¬S=¬((¬p∧q)∨(p∧¬q))
3. Use De Morgan's Law: According to De Morgan's laws, the negation of a disjunction is the conjunction of the negations:
¬S=¬(¬p∧q)∧¬(p∧¬q)
4. Apply De Morgan's Law to each part:
- For the first part ¬(¬p∧q):
¬(¬p∧q)=¬(¬p)∨¬(q)=p∨¬q
- For the second part ¬(p∧¬q):
¬(p∧¬q)=¬(p)∨¬(¬q)=¬p∨q
5. Combine the results:
¬S=(p∨¬q)∧(¬p∨q)
6. Final expression:
The negation of the original statement is: ¬S=(p∨¬q)∧(¬p∨q)
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