Problem 32 - A Complete Resource Book in Mathematics for JEE Main

Question

\(A\) set contains \((2 n+1)\) elements. The number of subsets of the set which contain at most \(n\) elements is (A) \(2^{n}\) (B) \(2^{n+1}\) (C) \(2^{2 n-1}\) (D) \(2^{2 n}\)

Step-by-Step Solution

Answer

(D) \(2^{2n}\)

Step 1: Understand the problem

We have a set with \((2n + 1)\) elements. We need to find the number of subsets that contain at most \(n\) elements.

Step 2: Compute total number of subsets

The total number of subsets of a set with \((2n + 1)\) elements is \(2^{2n + 1}\).

Step 3: Compute subsets with more than n elements

Subsets containing more than \(n\) elements have \(n+1, n+2, \ldots, 2n+1\) elements. By symmetry, the number of subsets with more than \(n\) elements equals those with less than or equal to \(n\) elements.

Step 4: Use symmetry to find the solution

Since the total number of subsets is \(2^{2n + 1}\), and due to symmetry, the number of subsets with more than \(n\) elements is equal to the number with at most \(n\) elements. Thus, the number of subsets with at most \(n\) elements is half of the total: \(2^{2n + 1} / 2 = 2^{2n}\).

Key Concepts

Subsets

In set theory, a subset is essentially any group or combination of elements that can be formed from an original set. Imagine you have a set of all the friends in your class - every possible smaller group of these friends is a subset. Here’s how the math works:

• The number of possible subsets of a given set equals \( 2^n \), where \( n \) is the number of elements in the original set.
• This relationship works because each element has two choices: either it's included in a subset or it isn't.

In our exercise, we deal with a set of \( (2n+1) \) elements. Calculating the total number of all subsets gives us \( 2^{2n+1} \), as each of those elements can either be included or excluded from a subset. Not all these subsets will be required, but knowing how many there are in total helps us find the subsets we do need.

Symmetry in Sets

Symmetry in sets is like finding balance. It helps to understand how different subsets can relate to each other. When the number of elements in a set is odd, like our \( (2n+1) \), "symmetry" plays a key role.
For every subset that contains more than half the elements of the set, there is a corresponding subset that contains fewer than half. This is because every "large" subset (more than \( n \) elements) can be paired with a "small" subset (less than \( n \) elements) whose elements are exactly those which the "large" subset excludes.
In the context of our problem,

• Subsets with more than \( n \) elements have \( n + 1, n + 2, \ldots, 2n + 1 \) elements.
• Number of these subsets is equal to the subsets with \( n \) or fewer elements.

Due to the symmetry, we realize that half of all subsets have at most \( n \) elements.

Set Theory

Set theory lays the foundation for understanding groups of items which could be numbers, objects, or even more abstract elements. It uses simple but deep concepts to generalize how we discuss things like subsets, union, intersection, and symmetry.
When approaching problems in set theory:

• Consider the total possible combinations (subsets).
• Think about how elements could be paired or separated, often leveraging ideas like symmetry.
• Apply mathematical operations to explore and figure out relationships between elements.

In our problem, set theory is applied to approach "at most" problems by counting subsets based on a condition (no more than half the elements). With principles like the number of subsets, symmetry, and dividing by considering the middle point of the set, set theory simplifies complicated problems into manageable parts. This allows us to determine that the number of subsets with at most \( n \) elements is actually half of the total number, thanks to the symmetrical nature of set elements and their combinations.