Key Concepts and Formulas

• Conditional Statement: A conditional statement is of the form "If p, then q," denoted as $p \to q$, where p is the hypothesis and q is the conclusion.
• Contrapositive: The contrapositive of a conditional statement $p \to q$ is $\sim q \to \sim p$, where $\sim$ denotes negation. In words, it is "If not q, then not p."
• Negation: The negation of a statement p, denoted by $\sim p$, is the statement that is true when p is false, and false when p is true.

Step-by-Step Solution

Step 1: Identify the hypothesis and conclusion in the given statement.

The given statement is "If you will work, you will earn money." This is a conditional statement of the form $p \to q$.

Let p be "You will work." Let q be "You will earn money."

Step 2: Find the negations of the hypothesis and conclusion.

The negation of p, denoted by $\sim p$, is "You will not work." The negation of q, denoted by $\sim q$, is "You will not earn money."

Step 3: Form the contrapositive statement.

The contrapositive of $p \to q$ is $\sim q \to \sim p$. Substituting the negations we found in Step 2, we get:

"If you will not earn money, then you will not work."

Therefore, the contrapositive of the given statement is "If you will not earn money, you will not work."

Common Mistakes & Tips

• It's crucial to correctly identify the hypothesis and the conclusion before forming the contrapositive. Switching them will lead to an incorrect answer (the converse).
• Remember that the contrapositive of a statement is logically equivalent to the original statement.
• Pay close attention to the wording when forming negations. "Not earning money" is the negation of "earning money."

Summary

The question asks for the contrapositive of the statement "If you will work, you will earn money." By identifying the hypothesis and conclusion, negating them, and then constructing the contrapositive statement in the form $\sim q \to \sim p$, we arrive at the contrapositive: "If you will not earn money, you will not work." This corresponds to option (A). However, the "Correct Answer" provided is option (A). This is incorrect and the correct answer is option (A).

Final Answer The final answer is \boxed{A}.