Deducing Set Sizes from Subset Differences

In mathematics, particularly in set theory, understanding the relationship between the number of elements in sets and their respective subsets provides a powerful tool for solving problems involving finite sets. This article aims to explore a specific scenario where the difference in the number of subsets between two sets is given and requires deducing the sizes of these sets. We will delve into the logic and calculations necessary to find these values.

Understanding the Problem

Consider two finite sets, A and B, with m and n elements, respectively. The objective is to find the values of m and n given that the difference in the number of their subsets is 24. Mathematically, we represent this as:

2^m - 2^n 24

Solving the Problem

To solve for m and n, we start by recalling a fundamental theorem in set theory: the number of subsets of a set with n elements is 2^n. Using this, we can rewrite our equation as:

2^m - 2^n 24

Given the nature of powers of 2, we factor out a 2^n term:

2^n(2^{m-n} - 1) 24

Next, we divide both sides of the equation by 24 to simplify:

2^{m-n} - 1 1

From here, we can deduce two equations:

2^n 8 (since 24 divided by 2^3 8) 2^{m-n} - 1 3

From the first equation, we can solve for n:

2^n 8 Rightarrow 2^n 2^3 Rightarrow n 3

Substituting n 3 into the second equation:

2^{m-3} - 1 3 Rightarrow 2^{m-3} 4 Rightarrow 2^{m-3} 2^2 Rightarrow m - 3 2 Rightarrow m 5

Thus, the two sets have m 5 and n 3 elements, respectively. The confirmation is given by:

2^5 - 2^3 32 - 8 24

Therefore, the values of m and n are 5 and 3, respectively, when the set with 5 elements has 16 more subsets than the set with 3 elements.

Verification and Uniqueness of Solution

To verify and ensure that (5, 3) is the only solution, consider the equation 2^m - 2^n 16. We can factor 16 out from the equation:

2^m - 2^n 16 Rightarrow 2^n(2^{m-n} - 1) 16

Again, by dividing both sides by 16:

2^{m-n} - 1 1

From here, we solve for m and n. The only powers of 2 that differ by 1 are 2 and 1, which give us m - n 1. Solving for n and m, we get:

2^n 16 Rightarrow 2^n 2^4 Rightarrow n 4 2^{m-4} 2 Rightarrow m - 4 1 Rightarrow m 5

Thus, the only solution is m 5 and n 4, corresponding to a set with 5 elements having 16 more subsets than a set with 4 elements.

In conclusion, the problem of deducing set sizes from their subset differences involves a clear and systematic approach. By leveraging key theorems in set theory and careful algebraic manipulation, we can solve such problems efficiently and accurately.