Question

Statement 1: Consider the statements:

  • p: Delhi is in India
  • q: Mumbai is not in Italy

Then the negation of the statement p OR q is "Delhi is not in India and Mumbai is in Italy."

Statement 2: For any two statements p and q, ~(p OR q) = ~p OR ~q.

Both Statement 1 and Statement 2 are false.

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Answer

To resolve the given statements and determine their correctness, let's analyze each one step-by-step:

Statement 1:

Given statements:

  • ( p ): Delhi is in India.
  • ( q ): Mumbai is not in Italy.

Negation Statement 1: We need to find the negation of ( p ) OR ( q ). The statement ( p ) OR ( q ) translates to: "Delhi is in India OR Mumbai is not in Italy."

To negate ( p ) OR ( q ), apply De Morgan's law, which states: $$ \neg (p , \text{OR} , q) = \neg p , \text{AND} , \neg q $$

Negated Statements:

  • ( \neg p ): Delhi is not in India.
  • ( \neg q ): Mumbai is in Italy.

Combined Negation: The negation of ( p ) OR ( q ) is: "Delhi is not in India and Mumbai is in Italy."

This matches what is stated in Statement 1. Therefore, Statement 1 is true.

Statement 2:

Given: For any two statements ( p ) and ( q ), $$ \neg (p , \text{OR} , q) = \neg p , \text{OR} , \neg q $$

Using De Morgan's law again, $$ \neg (p , \text{OR} , q) \neq \neg p , \text{OR} , \neg q $$ Instead, it is: $$ \neg (p , \text{OR} , q) = \neg p , \text{AND} , \neg q $$

Thus, Statement 2 is false as it contradicts De Morgan's law by stating disjunction (OR) instead of conjunction (AND).

Conclusion:

  • Statement 1 is true.
  • Statement 2 is false.

Thus, the given solution correctly identifies that both Statement 1 is true and Statement 2 is false.

Final Answer: C


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