The negation of the statement Aman or Ria lived in Lucknow " is

Options

A. Aman did not live in Lucknow and Ria lived in Lucknow
B. Aman lived in Lucknow and Ria did not live in Lucknow
C. Aman did not live in Lucknow and Ria did not live in Lucknow .
D. Aman did not live in Lucknow or Ria did not live Lucknow .

Answer

To find the negation of the statement "Aman or Ria lived in Lucknow," we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Original Statement**:
The original statement is "Aman or Ria lived in Lucknow." This can be represented as \( P: Q \lor R \), where:
- \( Q \) represents "Aman lived in Lucknow."
- \( R \) represents "Ria lived in Lucknow."

2. **Understand the Logical Structure**:
The statement uses the logical operator "or" (denoted as \( \lor \)). In logic, the statement \( Q \lor R \) is true if at least one of \( Q \) or \( R \) is true.

3. **Apply the Negation Rule**:
The negation of a disjunction (an "or" statement) is given by De Morgan's Laws. According to these laws:
\[ \neg(Q \lor R) = \neg Q \land \neg R \]
This means that the negation of "Aman or Ria lived in Lucknow" is equivalent to "Aman did not live in Lucknow and Ria did not live in Lucknow."

4. **Express the Negated Statement**:
Therefore, the negation of the statement can be written as:
- "Aman did not live in Lucknow" (which is \( \neg Q \))
- "Ria did not live in Lucknow" (which is \( \neg R \))

Combining these, we get:
\[ \text{Negation: "Aman did not live in Lucknow and Ria did not live in Lucknow."} \]

### Final Answer:
The negation of the statement "Aman or Ria lived in Lucknow" is:
**"Aman did not live in Lucknow and Ria did not live in Lucknow."**

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