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Question 73

The contrapositive of the statement "If you are born in India, then you are a citizen of India", is:

We begin by identifying the logical form of the given English statement. The sentence “If you are born in India, then you are a citizen of India” can be written symbolically as $$P \rightarrow Q,$$ where

$$P: \text{You are born in India},$$

$$Q: \text{You are a citizen of India}.$$

Now we recall the definition of a contrapositive. The logical rule states that for any conditional statement $$P \rightarrow Q,$$ the contrapositive is obtained by first negating both the hypothesis and the conclusion, and then reversing their order. In symbols,

$$\text{Contrapositive of } (P \rightarrow Q) \text{ is } (\lnot Q) \rightarrow (\lnot P).$$

So we apply this rule step by step. First, we negate the conclusion $$Q$$:

$$\lnot Q: \text{You are \emph{not} a citizen of India}.$$

Next, we negate the hypothesis $$P$$:

$$\lnot P: \text{You are \emph{not} born in India}.$$

Finally, we place the negated conclusion as the new hypothesis and the negated hypothesis as the new conclusion, giving

$$\lnot Q \rightarrow \lnot P,$$

which translates back into English as “If you are not a citizen of India, then you are not born in India.”

We now compare this derived contrapositive with the options provided:

A. If you are not born in India, then you are not a citizen of India.   (Matches $$\lnot P \rightarrow \lnot Q$$, which is the converse of the inverse, not the contrapositive.)

B. If you are a citizen of India, then you are born in India.   (This is the converse of the original statement.)

C. If you are born in India, then you are not a citizen of India.   (This directly contradicts the original statement.)

D. If you are not a citizen of India, then you are not born in India.   (Exactly matches $$\lnot Q \rightarrow \lnot P,$$ the contrapositive.)

We see that option D expresses the correct contrapositive.

Hence, the correct answer is Option D.

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