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.
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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