Key Concepts and Formulas

• Conditional Statement: A statement of the form "If p, then q", denoted as $p \rightarrow q$.
• Contrapositive: The contrapositive of the statement $p \rightarrow q$ is $\sim q \rightarrow \sim p$, where $\sim$ denotes negation. In words, the contrapositive of "If p, then q" is "If not q, then not p."
• Negation: The negation of a statement p, denoted by $\sim p$, is the statement that is true when p is false, and false when p is true.

Step-by-Step Solution

Step 1: Identify the component statements.

We are given the statement "If you are born in India, then you are a citizen of India." Let's identify the two component statements: $p$: You are born in India. $q$: You are a citizen of India. The given statement can be represented as $p \rightarrow q$.

Step 2: Find the negations of the component statements.

We need to find the negations of $p$ and $q$. $\sim p$: You are not born in India. $\sim q$: You are not a citizen of India.

Step 3: Form the contrapositive.

The contrapositive of $p \rightarrow q$ is $\sim q \rightarrow \sim p$. Substituting the negations we found in Step 2, we get: "If you are not a citizen of India, then you are not born in India."

Step 4: Compare with the given options.

The contrapositive we found is: "If you are not a citizen of India, then you are not born in India." This matches option (A).

Common Mistakes & Tips

• Remember the correct order for forming the contrapositive: negate both statements and reverse the implication. A common mistake is to only negate the statements or only reverse the implication.
• Pay close attention to the wording of the original statement and its components when forming the negations. For example, the negation of "all" is "not all" or "some are not", and the negation of "some" is "none".
• The statement and its contrapositive are logically equivalent, meaning they have the same truth value.

Summary

The contrapositive of a conditional statement "If p, then q" is formed by negating both p and q and reversing the direction of the implication, resulting in "If not q, then not p". In this problem, p is "You are born in India" and q is "You are a citizen of India." Therefore, the contrapositive of the given statement "If you are born in India, then you are a citizen of India" is "If you are not a citizen of India, then you are not born in India." This corresponds to option (A).

Final Answer The final answer is \boxed{A}.