Animated Solution for Mathematics - Sets and Relations: In a school, there are three types of games to be played. Some of the students play two types of games, but none play all the three games. Which Venn diagrams can justify the above statement?

A Venn Diagram Masterclass

Welcome, future engineers! Today, we are diving into a problem that tests not just your ability to calculate, but your ability to visualize logic. Venn diagrams are the bedrock of set theory, and in the high-stakes environment of the JEE, they are your most powerful tool for translating word problems into geometric reality.

Decoding the Language

The problem presents us with three games, which we can represent as sets $A$, $B$, and $C$. We are given two fundamental constraints:

1. Some students play two types of games. 2. None play all three games.

The first condition tells us that the pairwise intersections—$A\cap B$, $B\cap C$, and $A\cap C$—must be non-empty. This is our 'existence' condition.

The second condition, however, is our 'exclusion' condition. It tells us that the triple intersection, $A\cap B\cap C$, must be the empty set, $\varnothing$. In the world of sets, this is the 'Forbidden Zone'. If a diagram shows a region where all three circles overlap, it violates our second constraint.

The Visual Audit

Now, let's put our detective hats on and inspect the diagrams provided. We are looking for the 'Forbidden Zone'.

In Diagram P, we see two smaller circles nestled inside a larger one. Where they overlap, we see a clear, shared region. That region is the triple intersection, meaning $A \cap B \cap C eq \emptyset$. Therefore, Diagram P is disqualified.

Moving to Diagram Q, we see two circles side-by-side, with a third larger circle encompassing their intersection. Look at the center; the third circle covers the overlap of the first two. This creates a region common to all three, so the triple intersection is non-empty. Diagram Q is also disqualified.

Finally, let's examine Diagram R. Here, we have two large intersecting circles and a third smaller circle intersecting the boundary of the other two. If you trace the lines, you will find a small, distinct region where all three circles overlap. The triple intersection exists, and thus, Diagram R is disqualified.

The Final Verdict

We have analyzed all three diagrams, and in every single case, we found a non-empty triple intersection. The problem statement demands that no student plays all three games, which mathematically requires:

Since all three diagrams fail this condition, none of them can justify the statement. The correct answer is, quite elegantly, 'None of these'.

Remember, in JEE problems, the trap is often in the visual assumption. We assume a diagram must be correct, but sometimes, the beauty of the problem lies in realizing that the provided options simply do not fit the logical constraints. Keep your eyes sharp, trust your set theory, and never be afraid to conclude that 'none of the above' is the only logical path forward. You've got this!