The negative of the statement ''Rajesh or Rajni lived in Bengaluru'' is

Options

A. Rajes did not Live in Bangalur or Rajesh lives in Bangaluru.
B. Rajesh lives in Bangluru and Rajni lives in Bangalur
C. Rajesh did not live in Bangaluru and Rajn did not live in Bangaluru
D. Rajesh did not live in Bangaluru or Rajni did not live in Bangluru

Explanation

To find the negative of the statement "Rajesh or Rajni lived in Bengaluru," we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Original Statement**:
The original statement is "Rajesh or Rajni lived in Bengaluru." This can be expressed in logical terms as \( P \lor Q \), where \( P \) is "Rajesh lived in Bengaluru" and \( Q \) is "Rajni lived in Bengaluru."

2. **Understanding Negation**:
The negation of a disjunction (an "or" statement) is given by De Morgan's Laws. According to these laws, the negation of \( P \lor Q \) is \( \neg P \land \neg Q \). This means we need to negate both parts of the statement and change "or" to "and."

3. **Negate Each Part**:
- Negate \( P \): "Rajesh did not live in Bengaluru" becomes \( \neg P \).
- Negate \( Q \): "Rajni did not live in Bengaluru" becomes \( \neg Q \).

4. **Combine the Negated Statements**:
Now, we combine the negated parts using "and":
- The negation of the original statement is "Rajesh did not live in Bengaluru and Rajni did not live in Bengaluru."

5. **Final Statement**:
Therefore, the negative of the statement "Rajesh or Rajni lived in Bengaluru" is:
- "Rajesh did not live in Bengaluru and Rajni did not live in Bengaluru."

### Final Answer:
The negative of the statement "Rajesh or Rajni lived in Bengaluru" is:
"Rajesh did not live in Bengaluru and Rajni did not live in Bengaluru."

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