Exploring the Bounds of Altered Sets and Their Formulas

Set theory plays a fundamental role in understanding the behavior of various operations on sets, especially in the context of real numbers. This article delves into the properties of altered sets, specifically addressing the infimum and supremum of transformed sets. We will explore the key concepts and provide rigorous proofs for the given problems.

Basic Concepts and Definitions

Before we proceed, let's establish some basic concepts and definitions. Given a non-empty bounded subset A and B of the real numbers R, we consider the set C defined as C {3a - 2b : a€A, b€B}. We need to derive a formula for the infimum of C in terms of the infimum and supremum of A and B. Additionally, we will discuss the operations of scaling and multiplying subsets and their implications on their bounds.

Operations on Subsets and Their Bounds

We start by proving some general facts about the operations on subsets and their bounds:

Product of Subsets

Given two bounded subsets X and Y of R, we define the product set XY as {xy : x€X, y€Y}. We need to prove that the supremum and infimum of the product set XY are given by the product of the supremum and infimum of X and Y, respectively:

- sup(XY) sup(X)sup(Y)

- inf(XY) inf(X)inf(Y)

Scaling Subsets

Given a real number k, we define the scaled subset kX as {kx : x€X}. We need to prove that the supremum and infimum of the scaled subset kX are given by the product of k and the supremum and infimum of X respectively:

- If k 0, then sup(kX) ksup(X) and inf(kX) kinf(X)

- If k 0, then sup(kX) kinf(X) and inf(kX) ksup(X)

Proofs and Derivations

Proof for sup(XY) sup(X)sup(Y)

To prove that sup(XY) sup(X)sup(Y), we take the following steps:

If u sup(X) and v sup(Y), then uv is an upper bound for XY. Given any ε 0, there exist x€X and y€Y such that x u - ε/2 and y v - ε/2 Substituting these into the inequality, we get xy uv - ε. This shows that uv - ε is not an upper bound for XY, hence, uv is the least upper bound, or supremum, of XY.

The proof for the infimum is similar, and we leave it to the reader as an exercise to complete.

Proof for inf(XY) inf(X)inf(Y)

The proof follows a similar structure to the proof for the supremum:

Given u inf(X) and v inf(Y), uv is a lower bound for XY. For any ε 0, there exist x€X and y€Y such that x u ε/2 and y v ε/2. Therefore, xy uv ε. Hence, uv ε is not a lower bound for XY, implying that uv is the greatest lower bound, or infimum, of XY.

Application to the Given Problem

Now, let's apply these general facts to the problem at hand. Given C {3a - 2b : a€A, b€B}, we have:

- Let s sup(A) and u inf(B).

- The set C can be seen as a scaled and transformed version of the product set A(-2B).

Hence, by the general theorem for scaled subsets, we have:

- inf(3A - 2B) 3sup(A) - 2inf(B) 3s - 2u

Conclusion

This article has explored the bounds of altered sets and provided proofs for key properties. Understanding these concepts and theorems can help in various mathematical and real-world applications, such as optimization problems and economic modeling. The ability to manipulate and understand the behavior of sets under operations such as scaling and product sets is crucial in advanced mathematics and its applications.

References

[1] Rudin, W. (1976). Principles of Mathematical Analysis. McGraw-Hill.

[2] Johnsonbaugh, R. (1997). Foundations of Mathematical Analysis. Prentice Hall.