Sets, Relations & Functions
Mathematical Reasoning
Grade 11
Question:
<p>Consider the statement \(p\): 'New Delhi is a city'. Which of the following is not negation of \(p\)?</p>
<p>New Delhi is not a city</p>
<p>It is false that New Delhi is a city</p>
<p>It is not the case that New Delhi is a city</p>
<p>None of these</p>
Step-by-Step Solution
Key Concept: The negation of a statement p must be logically opposite and make p false. A statement that is merely different or unrelated to p is not its negation—negation requires a direct contradiction.
<p><strong>Step 1:</strong> Identify statement p: 'New Delhi is a city' (TRUE statement)</p><p><strong>Step 2:</strong> Determine what negation of p must be: ¬p = 'New Delhi is not a city' (This makes p false and is the direct logical opposite)</p><p><strong>Step 3:</strong> The negation must:<br/>• Directly contradict p<br/>• Be false when p is true<br/>• Express the opposite meaning</p><p><strong>Step 4:</strong> Evaluate each option:<br/>• A statement that says 'New Delhi is not a city' = IS a negation<br/>• A statement that says 'It is false that New Delhi is a city' = IS a negation<br/>• A statement that says 'Delhi is the capital of India' = NOT a negation (it's a different true statement, not a contradiction of p)<br/>• A statement denying p's truth = IS a negation</p><p><strong>Step 5:</strong> The option that is unrelated to p or simply a different statement (not its logical opposite) is NOT a negation.</p><p>∴ Answer: D (The specific option that doesn't directly contradict p)</p>
Correct Answer: D