Animated Solution for Mathematics - Basic Mathematics: Contrapositive of the statement "If two numbers are not equal, then their squares are not equal." Is:

Welcome, future engineer. Today, we are not just solving a problem; we are peeling back the layers of human thought to understand the very foundation of mathematical reasoning: Logic.

Often, students rush to calculate, but in logic, the beauty lies in the structure. Let us dissect the statement: "If two numbers are not equal, then their squares are not equal."

To master this, we must first identify the skeleton. Every conditional statement follows the form "If $p$, then $q$".

Here, our antecedent $p$ is "two numbers are not equal", which we can write as $x eq y$. Our consequent $q$ is "their squares are not equal", which we write as $x^2 eq y^2$.

So, our original statement is simply $p\to q$.

Now, the question asks for the contrapositive. This is a specific logical operation. Many students confuse it with the converse or the inverse.

Let us be precise. The contrapositive of $p\to q$ is defined as $\sim q\to \sim p$.

This is not just a rearrangement; it is a transformation that preserves the logical integrity of the original statement. It is a powerful tool in proofs because it allows us to approach a problem from a different angle while maintaining the same truth value.

To find the contrapositive, we must perform two steps: negate the consequent and negate the antecedent.

First, let us look at $q$: "$x^2 eq y^2$". The negation, $\sim q$, is the exact opposite: "$x^2=y^2$".

Next, we look at $p$: "$x eq y$". The negation, $\sim p$, is "$x=y$". We have now successfully identified our building blocks.

The Final Statement

Now, we assemble them into the structure $\sim q\to \sim p$. We start with $\sim q$: "If the squares of two numbers are equal".

Then, we follow with $\sim p$: "then the numbers are equal". Putting it together, we get:

This is the contrapositive. It is elegant, it is precise, and it is logically equivalent to the original statement. By mastering this simple "flip and negate" rule, you have unlocked a fundamental technique used in everything from basic algebra to advanced real analysis.