Problem 58 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The contrapositive of the statement "If it is raining, then I will not come", is: (a) IfI will not come, then it is raining. (b) If I will not come, then it is not raining. (c) IfI will come, then it is raining. (d) If I will come, then it is not raining.

Step-by-Step Solution

Answer

The contrapositive is: (d) If I will come, then it is not raining.

Step 1: Identify the Original Statement Components

In logic, an implication is generally expressed in the form "If P, then Q", where P is the hypothesis and Q is the conclusion. In this exercise, the original statement is "If it is raining (P), then I will not come (Q)".

Step 2: Understand the Contrapositive Concept

The contrapositive of a statement "If P, then Q" is "If not Q, then not P". The contrapositive is logically equivalent to the original statement, meaning it is always true if the original statement is true.

Step 3: Apply Contrapositive to the Original Statement

Using the original statement "If it is raining, then I will not come", we: - Identify P: "It is raining" - Identify Q: "I will not come" The contrapositive changes this to "If it is not true that I will not come, then it is not raining".

Step 4: Simplify the Contrapositive Statement

The contrapositive "If it is not true that I will not come, then it is not raining" can be simplified to "If I will come, then it is not raining".

Step 5: Match the Contrapositive to Options Provided

Now we compare the simplified contrapositive statement "If I will come, then it is not raining" to the provided options. We find that option (d) "If I will come, then it is not raining" matches the contrapositive statement.

Key Concepts

Contrapositive Statement

In logic, understanding what a contrapositive statement is can help you evaluate logical arguments more effectively. A contrapositive statement involves taking an original conditional statement and transforming it. Suppose we have a statement "If P, then Q" where P is the hypothesis and Q is the conclusion. The contrapositive of this statement is "If not Q, then not P." This transformation involves two main steps:

• First, negate both the hypothesis and the conclusion from the original statement.
• Second, switch their positions.

Consider the example "If it is raining (P), then I will not come (Q)." Its contrapositive is "If I will come, then it is not raining." The contrapositive is logically equivalent to the original statement, meaning they have the same truth value. Whenever the original is true, the contrapositive is true, and vice versa.

Hypothesis and Conclusion

Identifying the hypothesis and conclusion is fundamental when working with logical statements. Typically, these are framed in a format "If P, then Q," where:

• P represents the hypothesis.
• Q represents the conclusion.

In the statement "If it is raining, then I will not come," "It is raining" is the hypothesis, and "I will not come" is the conclusion. In any logically sound argument or mathematical proof, correctly identifying these components helps clarify the argument's structure and ensure accuracy in results. Misinterpreting these can lead to errors, highlighting the importance of distinguishing between what is assumed (hypothesis) and what follows (conclusion).

Logical Equivalence

Logical equivalence is a key concept in logical reasoning. Two statements are said to be logically equivalent if, in every possible situation, they have the same truth value. In simpler terms, they are either both true or both false. A classic example of logical equivalence is between a statement and its contrapositive. As introduced earlier, the contrapositive "If not Q, then not P" of the original conditional "If P, then Q" maintains logical equivalence. For instance, both "If it is raining, then I will not come" and its contrapositive "If I will come, then it is not raining" are equivalent. Understanding logical equivalence helps in:

• Verifying proofs and arguments.
• Simplifying complex logical expressions.
• Ensuring consistency in logical deductions.

By mastering logical equivalences, you become adept at rearranging logical statements without altering their inherent truth, enabling a deeper grasp of logical concepts.