Question

Negation of \( p \land (q \land \neg (p \land q)) \) is:}

• A. ~(p∨q)
• B. p∨q
• C. (~(p∧q))∧q
• D. (~(p∧q))∨p

Answer 1

The Correct Option is D

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)