Animated Solution for Mathematics - Basic Mathematics: The contrapositive of the statement "If you will work, you will earn money" is:

The statement "If you will work, you will earn money" serves as a foundational implication in symbolic logic. We represent this as $p\to q$, where $p$ is the hypothesis "You will work" and $q$ is the conclusion "You will earn money."

This structure acts as a contract between effort and reward, forming the bedrock of our logical analysis.

To explore the implication of not earning money, we utilize the contrapositive. This is a fundamental logical transformation defined by the rule:

To derive this, we perform two distinct operations: we swap the hypothesis and the conclusion, and we negate both components.

First, we identify the negations of our original components. Since $p$ is "You will work," its negation $\sim p$ is "You will not work."

Similarly, since $q$ is "You will earn money," its negation $\sim q$ is "You will not earn money."

By applying the rule $\sim q\to \sim p$, we arrive at the final statement: "If you will not earn money, you will not work."

This transformation is more than a mere rearrangement of words; it is a profound logical truth. A statement and its contrapositive are logically equivalent, meaning they possess the exact same truth value.

If the original promise holds, the contrapositive must necessarily hold as well. This precision is the hallmark of mathematical logic, allowing us to navigate complex arguments with absolute confidence.

As you prepare for the JEE, remember that mastering these transformations is essential for sharpening your analytical mind. Logic is the language of the universe, and understanding its structure is the key to unlocking the most challenging problems.