Animated Solution for Mathematics - Basic Mathematics: The contrapositive of the statement "If I reach the station in time, then I will catch the train" is:

We are given the statement: "If I reach the station in time, then I will catch the train." In the language of logic, this is a conditional statement.

We can represent it symbolically as $P\to Q$. Here, $P$ is the condition: "I reach the station in time", and $Q$ is the result: "I will catch the train".

The arrow $\to$ represents the logical implication. It tells us that if $P$ happens, $Q$ is guaranteed to follow. It is a simple, beautiful flow of causality.

The question asks for the contrapositive. This is where many students stumble, but you won't!

The contrapositive of $P\to Q$ is defined as $\sim Q\to \sim P$. The symbol $\sim$ denotes negation, or "NOT".

So, we are saying: "If NOT $Q$, then NOT $P$". To find the contrapositive, we perform two steps: 1. Negate both propositions. 2. Swap their positions.

Let us apply this to our statement. Our $Q$ is "I will catch the train". Its negation, $\sim Q$, is "I will not catch the train".

Our $P$ is "I reach the station in time". Its negation, $\sim P$, is "I do not reach the station in time".

Now, we put them together in the form $\sim Q\to \sim P$: "If I will not catch the train, then I do not reach the station in time."

You might wonder, why do we do this? Why not just swap them?

If we just swapped them, we would get the converse, which is not logically equivalent. The beauty of the contrapositive is that it is logically equivalent to the original statement.

If the original statement is true, the contrapositive must also be true. It is a mirror image that preserves the truth value.

In the exam, you will encounter many such logical statements. The key is to stay calm, define your $P$ and $Q$ clearly, and apply the transformation rule systematically.

Do not let the English language confuse you; translate it into the symbolic language of logic, and the answer will reveal itself.

You have the tools, you have the logic, and now you have the understanding. Go forth and conquer these problems with confidence!