Key Concepts and Formulas

• Conditional Statement: A statement of the form "If p, then q," denoted as $p \rightarrow q$.
• Contrapositive: The contrapositive of the conditional statement $p \rightarrow q$ is $\sim q \rightarrow \sim p$, where $\sim p$ means "not p". In other words, the contrapositive negates both the hypothesis (p) and the conclusion (q) and reverses their order.

Step-by-Step Solution

Step 1: Identify the component statements

Let's break down the given statement into its component parts:

$p$: I reach the station in time. $q$: I will catch the train.

The given statement "If I reach the station in time, then I will catch the train" can be represented as $p \rightarrow q$.

Step 2: Find the negations of the component statements

Now, we need to find the negations of $p$ and $q$:

$\sim p$: I do not reach the station in time. $\sim q$: I will not catch the train.

Step 3: Form the contrapositive statement

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

"If I will not catch the train, then I do not reach the station in time."

Step 4: Compare with the given options

Comparing the contrapositive statement we derived with the given options, we see that it matches option (C).

Common Mistakes & Tips

• Confusing Contrapositive with Converse/Inverse: Remember that the contrapositive is formed by both negating and reversing the hypothesis and conclusion. The converse ($q \rightarrow p$) and inverse ($\sim p \rightarrow \sim q$) are different and have different truth values compared to the original statement.
• Negating Statements Correctly: Pay close attention to how you negate statements. For example, the negation of "I reach the station in time" is "I do not reach the station in time."
• Understanding Logical Equivalence: A statement and its contrapositive are logically equivalent, meaning they always have the same truth value.

Summary

To find the contrapositive of the given statement, we first identified the hypothesis and conclusion. Then, we negated both the hypothesis and the conclusion and reversed their order to form the contrapositive. This resulted in the statement "If I will not catch the train, then I do not reach the station in time," which corresponds to option (C).

Final Answer

The final answer is \boxed{C}, which corresponds to option (C).