Problem 51 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The contrapositive of the statement "If you are born in India, then you are a citizen of India", is : (a) If you are not a citizen of India, then you are not born in India. (b) If you are a citizen of India, then you are born in India. (c) If you are born in India, then you are not a citizen of India. (d) If you are not born in India, then you are not a citizen of India.

Step-by-Step Solution

Answer

(a) If you are not a citizen of India, then you are not born in India.

Step 1: Understand the Given Statement

The given statement is: "If you are born in India, then you are a citizen of India." This is an implication statement and can be expressed as: If \( p \), then \( q \), where \( p \) is "you are born in India" and \( q \) is "you are a citizen of India."

Step 2: Define the Contrapositive

The contrapositive of an implication statement "If \( p \), then \( q \)" is formulated as "If not \( q \), then not \( p \)." So, we need to identify the contrapositive of the given statement.

Step 3: Determine the Contrapositive

For the statement "If you are born in India, then you are a citizen of India," the contrapositive will be: "If you are not a citizen of India, then you are not born in India." Here, not \( q \) is "not a citizen of India" and not \( p \) is "not born in India."

Step 4: Match with Options Provided

We compare our determined contrapositive "If you are not a citizen of India, then you are not born in India" with the provided options. - (a) If you are not a citizen of India, then you are not born in India. - (b) If you are a citizen of India, then you are born in India. - (c) If you are born in India, then you are not a citizen of India. - (d) If you are not born in India, then you are not a citizen of India. The correct match is option (a).

Key Concepts

Implication in Logic

Implications are a fundamental part of logical reasoning. In simple terms, an implication indicates a condition and a result. You can think of it as a cause-effect relationship.
This is often expressed in the form "If...then...". For example, "If you study hard, then you will pass the exam". Here, studying hard is the condition, and passing the exam is the result.
In logic, the implication is expressed as "If \( p \), then \( q \)", where \( p \) is a premise and \( q \) is a conclusion. This forms the foundation for evaluating and constructing arguments.

Logical Reasoning

Logical reasoning is the process of using a structured approach to arrive at valid conclusions. When we use logical reasoning, we rely on the information given, follow a sequence of statements, and apply rules to ensure our conclusions are sound.
Logical reasoning is used across different fields such as mathematics, computer science, and everyday decision making. It helps us assess arguments, identify errors in reasoning, and build articulate arguments.

• Analyzing statements: Assessing whether statements are true.
• Constructing arguments: Putting together a series of logical steps to support a conclusion.
• Evaluating arguments: Checking the logical consistency and validity of conclusions.

By refining logical skills, one can improve problem-solving and critical thinking abilities.

Conditional Statements

Conditional statements are the backbone of logic and reasoning. They are statements that are made up of two parts: a hypothesis and a conclusion, in the form "If...then...". For example, "If it rains, then the ground will be wet."
These statements are powerful tools for making logical inferences. They allow us to understand the relationship between different propositions. In logic, conditional statements can be used to form contrapositive, converse, and inverse forms.

• Contrapositive: Swapping and negating both parts. The contrapositive of "If \( p \), then \( q \)" is "If not \( q \), then not \( p \)".
• Converse: Swapping the hypothesis and conclusion. The converse of "If \( p \), then \( q \)" is "If \( q \), then \( p \)".
• Inverse: Negating both parts. The inverse of "If \( p \), then \( q \)" is "If not \( p \), then not \( q \)".

Understanding these forms allows us to evaluate which statements are logically equivalent or lead to valid conclusions.