The contrapositive of a conditional statement is a fundamental concept in logic. A conditional statement has the form "If P, then Q", written as $P\to Q$. The contrapositive of this statement is "If not Q, then not P", written as $\sim Q\to \sim P$.

A key property is that a conditional statement and its contrapositive are logically equivalent. This means they always have the same truth value; if the original statement is true, the contrapositive is also true, and vice versa.

Let's apply this to the given statement: "If you are born in India, then you are a citizen of India."

Step 1: Identify P and Q.
P: You are born in India.
Q: You are a citizen of India.

Step 2: Form the contrapositive: "If not Q, then not P."
Not Q: You are not a citizen of India.
Not P: You are not born in India.

Therefore, the contrapositive is: "If you are not a citizen of India, then you are not born in India."

Comparing this with the given options, the correct one is: "If you are not a citizen of India, then you are not born in India."

Related Concepts

Converse: The converse of a conditional statement $P\to Q$ is $Q\to P$ ("If Q, then P"). The converse is not logically equivalent to the original statement. For our example, the converse would be "If you are a citizen of India, then you are born in India," which is a different (and in this specific real-world case, incorrect) statement.

Inverse: The inverse of a conditional statement $P\to Q$ is $\sim P\to \sim Q$ ("If not P, then not Q"). Like the converse, the inverse is also not logically equivalent to the original statement.

Key Formulae

Original Statement: $P\to Q$

Contrapositive: $\sim Q\to \sim P$ (Logically Equivalent)

Converse: $Q\to P$ (Not Logically Equivalent)

Inverse: $\sim P\to \sim Q$ (Not Logically Equivalent)