Problem 60 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The contrapositive of the statement "I go to school if it does not rain" is (a) If it rains, I do not go to school. (b) If I do not go to school, it rains. (c) If it rains, I go to school. (d) If I go to school, it rains.

Step-by-Step Solution

Answer

The contrapositive is option (b): "If I do not go to school, it rains."

Step 1: Understand the Conditional Statement

The given statement "I go to school if it does not rain" can be expressed in a conditional form: "If it does not rain, then I go to school." This follows the logical form: \(P \rightarrow Q\), where \(P\) is "it does not rain" and \(Q\) is "I go to school."

Step 2: Identify the Contrapositive

The contrapositive of a conditional statement \(P \rightarrow Q\) is \(eg Q \rightarrow eg P\), where \(eg Q\) is the negation of \(Q\) and \(eg P\) is the negation of \(P\). In this case, \(eg Q\) is "I do not go to school" and \(eg P\) is "it rains." So, the contrapositive is: "If I do not go to school, then it rains."

Step 3: Match the Contrapositive with Given Options

Now, we need to match the contrapositive "If I do not go to school, then it rains" with the given options. Comparing it to the provided choices: (a) If it rains, I do not go to school. (b) If I do not go to school, it rains. (c) If it rains, I go to school. (d) If I go to school, it rains. Upon comparison, option (b) "If I do not go to school, it rains" is exactly the contrapositive of the given statement.

Key Concepts

Understanding the Contrapositive

In the realm of logic, understanding the contrapositive is key to mastering logical reasoning. Contrapositive transformations are a pivotal part of logical proofs and are often used in mathematical reasoning.The contrapositive of a statement reverses and negates both the hypothesis and conclusion of a conditional statement.To illustrate, let's consider the conditional statement:- "If it does not rain, then I go to school."This statement matches the standard form: - If \(P\), then \(Q\), where \(P\) is "it does not rain," and \(Q\) is "I go to school."The contrapositive swaps and negates \(P\) and \(Q\), giving:- If \(eg Q\), then \(eg P\), or "If I do not go to school, then it rains."This transformation is logically equivalent to the original statement, a concept that is both powerful and often counterintuitive in logical reasoning.This means, in terms of truth value, the contrapositive and the original statement will always align.

Conditional Statements in Logic

Conditional statements serve as the backbone for logical reasoning and proofs.They are structured in an "If...then..." format, where one event leads to another.In logical terms, the standard form of a conditional statement is \(P \rightarrow Q\).Let's break down a real-world example:- Consider "I will go to school if it does not rain." This translates to "If it does not rain, then I will go to school."Here, "not raining" is the condition for "going to school."In conditional statements:- \(P\) represents the hypothesis ("it does not rain")- \(Q\) is the conclusion ("I go to school")Understanding how to structure these statements clearly and concisely is crucial to solving logical problems.These statements allow us to form predictions and suit various scenarios in our daily problem-solving tasks.

Logic and Proofs

Logic and proofs form the foundation for mathematical reasoning, allowing us to conclude with certainty. These concepts involve creating a series of logical steps to arrive at a conclusion, ensuring that each argument follows from its predecessor. A central component of logical proofs is deductive reasoning, where: - Arguments are formed from general premises to a specific conclusion. - Proofs rely on a solid structure where each step is justified logically. When working on proofs, different types of logical statements such as conditional, converse, inverse, and contrapositive play important roles: - The contrapositive is particularly valuable because of its equivalence to the conditional statement. - This equivalence allows for alternate forms of the same logical argument, aiding in proof creation. In practice, mastering these concepts helps students not just in mathematics but in developing critical thinking skills essential for diverse fields of study and problem-solving scenarios.