`"The negation of the statement= " (p vv q)^^ r` is

Options

A. `(~p vv~q) vv ~q`
B. `(~p ^^ ~q) vv ~r`
C. `~(p vv q) rarr r`
D. `p^^q`.

Answer

To find the negation of the statement \( (p \lor q) \land r \), we will use De Morgan's laws and the rules of negation in logic. Here’s how to do it step by step:

### Step 1: Write down the original statement
The original statement is:
\[ (p \lor q) \land r \]

### Step 2: Apply negation to the entire statement
To negate the statement, we write:
\[ \neg((p \lor q) \land r) \]

### Step 3: Apply De Morgan's Law
According to De Morgan's laws, the negation of a conjunction (AND) is the disjunction (OR) of the negations. Therefore, we can rewrite the negation as:
\[ \neg(p \lor q) \lor \neg r \]

### Step 4: Apply De Morgan's Law again to the first part
Now we need to negate \( (p \lor q) \). Again using De Morgan's laws, we have:
\[ \neg(p \lor q) = \neg p \land \neg q \]

### Step 5: Substitute back into the expression
Now we substitute this back into our expression:
\[ \neg(p \lor q) \lor \neg r = (\neg p \land \neg q) \lor \neg r \]

### Step 6: Final expression
Thus, the negation of the original statement \( (p \lor q) \land r \) is:
\[ (\neg p \land \neg q) \lor \neg r \]

### Summary
The negation of the statement \( (p \lor q) \land r \) is \( (\neg p \land \neg q) \lor \neg r \).

---