Problem 46 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The contrapositive of the statement "If \(I\) reach the station in time, then \(I\) will catch the train" is: \(\quad\) (a) If \(I\) do not reach the station in time, then \(I\) will catch the train. (b) If \(I\) do not reach the station in time, then \(I\) will not catch the train. (c) If \(I\) will catch the train, then \(I\) reach the station in time. (d) If \(I\) will not catch the train, then \(I\) do not reach the station in time.

Step-by-Step Solution

Answer

(d) If I will not catch the train, then I do not reach the station in time.

Step 1: Understand the Original Statement

The original statement is: "If \( I \) reach the station in time, then \( I \) will catch the train." This is a conditional statement, usually written as "If \( p \), then \( q \)," where \( p \) is "I reach the station in time" and \( q \) is "I will catch the train."

Step 2: Define the Contrapositive

The contrapositive of a statement "If \( p \), then \( q \)" is "If not \( q \), then not \( p \)." It involves negating both the hypothesis and the conclusion and then switching them.

Step 3: Apply the Contrapositive Definition

Negate both parts of the original statement. Not \( q \) becomes "I will not catch the train." Not \( p \) becomes "I do not reach the station in time." Thus, the contrapositive is: "If \( I \) will not catch the train, then \( I \) do not reach the station in time."

Step 4: Select the Correct Option

The correct statement that represents the contrapositive is: "If \( I \) will not catch the train, then \( I \) do not reach the station in time," which corresponds to option (d).

Key Concepts

Understanding Conditional Statements

A conditional statement is a logical assertion that connects two statements, typically expressed as "If \( p \), then \( q \)."
This type of statement is fundamental in mathematical logic because it establishes a relationship between two propositions, whereby the truth of one statement guarantees the truth of the second.
In our example, "If I reach the station in time, then I will catch the train," the condition of reaching the station in time (\( p \)) must be met for the subsequent event of catching the train (\( q \)) to occur.

Key features of a conditional statement include:

• Hypothesis (\( p \)): The part after 'If' which is the condition.
• Conclusion (\( q \)): The outcome that follows if the condition holds.

Exploring Negation in Logic

In logic, negation refers to the process of converting a statement to express the opposite meaning. This is done by introducing 'not' to change the truth value of the given proposition.
Negation plays a crucial role in forming a contrapositive. In a contrapositive, both the hypothesis and conclusion of the conditional statement are negated and reversed.

For instance, consider the statement "I will catch the train". Its negation is "I will not catch the train." Similarly, the negation of "I reach the station in time" is "I do not reach the station in time."

Negation helps dismantle and reconstruct statements to examine their truth from an alternative perspective. It provides insight into how altering one part of a statement affects the overall logical structure.

The Role of Logical Reasoning

Logical reasoning is the process of using a structured, disciplined approach to arrive at a conclusion. When dealing with conditional statements, logical reasoning helps us understand the implications of each statement, including its contrapositive or converse.
By evaluating these different forms, we can explore all possible logical outcomes and understand their interdependent nature.

In the exercise, to determine the contrapositive, we used logical reasoning to:

• Negate both the hypothesis and conclusion.
• Switch them to reflect the contrapositive syntax.

This methodical approach ensures the accuracy of our interpretation and enhances our ability to apply logic in various scenarios.

Introduction to Mathematical Logic

Mathematical logic is a branch of mathematics exploring formal systems, which are frameworks for expressing logical statements and performing logical operations. It provides tools and principles to frame statements logically and derive conclusions accurately.

Key principles in mathematical logic include conditional statements, their contrapositive, converse, and the process of negation. By using mathematical logic, we can solve problems systematically, ensuring our reasoning is sound and our conclusions are valid.

In our exercise, mathematical logic allowed us to transform the original statement into its contrapositive, verifying both the structure and truth value of derived statements. By understanding and applying these logical frameworks, we ensure that our problem-solving methods are not only correct but universally applicable.