Animated Solution for Mathematics - Sets and Relations: Let A, B, C and D be four non-empty sets. The Contrapositive statement of “If $A\subseteq B$ and $B\subseteq D$ then $A\subseteq C$” is :

Unlocking the Contrapositive

Welcome, fellow explorer of mathematical structures! Today, we are going to dissect a classic problem in set theory and logic.

It might seem like a simple exercise in rearranging words, but beneath the surface lies the bedrock of rigorous mathematical thinking. We are tasked with finding the contrapositive of the statement: "If $A\subseteq B$ and $B\subseteq D$, then $A\subseteq C$."

Deconstructing the Statement

First, let us look at the structure. We have a conditional statement, which we can represent as $P\Rightarrow Q$.

Here, the premise $P$ is actually a compound statement: "$A\subseteq B$ and $B\subseteq D$". The conclusion $Q$ is "$A\subseteq C$".

To make this manageable, let us assign logical variables to these components: - Let $p$ be the statement: $A\subseteq B$ - Let $q$ be the statement: $B\subseteq D$ - Let $r$ be the statement: $A\subseteq C$

Our original statement now takes the elegant symbolic form: $(p\wedge q)\Rightarrow r$. By stripping away the English, we can see the skeleton of the logic clearly.

The Contrapositive Dance

Now, we invoke the rule of the contrapositive. The contrapositive of any implication $X\Rightarrow Y$ is defined as $ eg Y \Rightarrow eg X$.

This is not just a rule; it is a logical identity. It means that the original statement and its contrapositive are always true or false together.

Applying this to our symbolic form, we get:

This is where the journey gets exciting. We have successfully reversed the direction of the implication and negated both sides. But we are not done yet!

The De Morgan's Trap

Here is where many students stumble. We have the term $ eg(p \land q)$. How do we expand this?

We must use De Morgan's Law, a powerful tool that governs how negations interact with conjunctions and disjunctions. De Morgan's Law tells us that the negation of a conjunction is the disjunction of the negations: $ eg(p \land q) \equiv ( eg p \lor eg q)$.

Notice the beautiful symmetry here: the "and" ($\wedge$) flips into an "or" ($\vee$). If you forget to flip that operator, you fall into the trap!

Substituting this back into our contrapositive expression, we arrive at:

Now, we simply translate our symbolic logic back into the language of sets: - $ eg r$ becomes $A ot\subseteq C$ - $ eg p$ becomes $A ot\subseteq B$ - $ eg q$ becomes $B ot\subseteq D$

Putting it all together, our final contrapositive statement is: "If $A ot\subseteq C$, then $A ot\subseteq B$ or $B ot\subseteq D$."

This matches perfectly with our fourth option. You have just navigated the labyrinth of logical implication, De Morgan's Law, and set theory. Keep this precision in your toolkit—it is exactly what separates the good from the great in JEE Advanced!