Question

Let \(S\) and \(T\) be nonempty sets of real numbers such that every real number is in \(S\) or \(T\) and if \(s \in S\) and \(t \in T,\) then \(s

Short answer

Question: Prove Dedekind's theorem which states that for two nonempty sets of real numbers, S and T, with the property that for every s in S and t in T, s < t, there exists a unique real number β such that every real number less than β is in S and every real number greater than β is in T. Answer: Dedekind's theorem is proven using the completeness property of real numbers, which ensures the existence of a least upper bound (supremum) for nonempty, bounded sets. We establish the existence of a unique real number β that separates set S and set T by first demonstrating its existence through the supremum and then showing its uniqueness by contradiction.

Step-by-step solution

Existence of \(\beta\)

To show the existence of \(\beta\), we will use the completeness property of real numbers, which states that if a nonempty set of real numbers has an upper bound, then it has a least upper bound, also known as a supremum. We know that set \(S\) is nonempty and has an upper bound. Since any element of set \(T\) is greater than every element of set \(S\), we can take any element from set \(T\) as an upper bound for set \(S\). Let's denote this upper bound by \(t' \in T\). Therefore, set \(S\) is bounded above by \(t'\). Using the completeness property, we can say that set \(S\) has a least upper bound (supremum). Let this least upper bound be \(\beta\). Then, \(\beta\) has the property that it is the smallest real number such that all real numbers less than \(\beta\) belong to set \(S\). Now, since every real number belongs to either set \(S\) or set \(T\), all the real numbers greater than \(\beta\) should belong to set \(T\). Hence, every real number less than \(\beta\) is in set \(S\) and every real number greater than \(\beta\) is in set \(T\).

Uniqueness of \(\beta\)

To show that the real number \(\beta\) is unique, we will assume the existence of another real number \(\beta'\) such that every real number less than \(\beta'\) is in set \(S\) and every real number greater than \(\beta'\) is in set \(T\). We will show that this leads to a contradiction, proving our assumption wrong and hence proving the uniqueness of \(\beta\). If \(\beta' > \beta\), then there exists real numbers between \(\beta\) and \(\beta'\). Since all real numbers belong to either set \(S\) or set \(T\), any number between \(\beta\) and \(\beta'\) should also belong to set \(S\) or set \(T\). However, this contradicts the property of \(\beta\) because there should not be any real numbers between \(\beta\) and any element of set \(S\). On the other hand, if \(\beta' < \beta\), then there exists real numbers between \(\beta\) and \(\beta'\). Since all real numbers belong to either set \(S\) or set \(T\), any number between \(\beta'\) and \(\beta\) should also belong to set \(S\) or set \(T\). However, this contradicts the property of \(\beta'\) that all real numbers less than \(\beta'\) belong to set \(S\) and that \(\beta\) is the least upper bound. In either case, we reach a contradiction which shows that our assumption of another real number \(\beta'\) with the same properties as \(\beta\) is wrong. Thus, the real number \(\beta\) is unique. In conclusion, there exists a unique real number \(\beta\) such that every real number less than \(\beta\) is in \(S\) and every real number greater than \(\beta\) is in \(T\), proving Dedekind's theorem.

Key concepts

Least Upper Bound

Understanding the concept of the least upper bound is pivotal when studying real numbers. In simple terms, the least upper bound, often abbreviated as LUB or written as supremum, is the smallest value that is greater than or equal to every number in a set. Imagine you have a set of numbers that do not surpass a certain point; the least upper bound will be the 'ceiling' that none of these numbers can exceed.

For example, in a set of numbers less than 3, the number 3 is not part of the set but serves as the least upper bound because it's the smallest number that no member of the set exceeds. This concept is essential in demonstrating the existence of a specific boundary in sets of real numbers, like set \(S\), in Dedekind's theorem.

Real Numbers

Real numbers form a continuous and complete set of numbers including all the rational numbers (like 1, 1/2, -3, and so on) and all the irrational numbers (like \(\sqrt{2}\), \(\pi\), etc.). What makes real numbers fascinating is their ability to represent any quantity along the number line, be it lengths, weights, or other measurements.

Because of this continuity, within any two distinct real numbers, we can find another real number – making the set dense. The problem in Dedekind's theorem revolves around splitting these densely packed numbers into two distinct sets \(S\) and \(T\), a concept that relies heavily on our understanding of real numbers.

Completeness Property

The completeness property of real numbers is a foundational characteristic that distinguishes them from other number sets like rational or natural numbers. This property ensures that every non-empty set of real numbers that is bounded above has a least upper bound. Similarly, every non-empty set that is bounded below has a greatest lower bound.So, the exercise showcases this property by affirming that set \(S\) has a supremum because it is assuredly non-empty and has an upper bound provided by the elements of set \(T\). The completeness property reassures us that \(\beta\), the least upper bound of \(S\), exists and lies within the real numbers.

Dedekind Cut

A Dedekind cut is a sophisticated way of slicing the real numbers into two distinct, non-overlapping sets. It is named after the German mathematician Richard Dedekind, who first formulated the concept. A Dedekind cut involves dividing the set of all real numbers into a lower set \(S\) and an upper set \(T\), such that every element in \(S\) is less than every element in \(T\), and \(S\) contains all elements less than a certain number, while \(T\) contains all those greater than it.

Significance in Real Analysis

Through Dedekind cuts, we can understand the continuity of real numbers and verify the presence of irrational numbers between rationals, filling in the 'gaps' and making our system complete. The exercise uses Dedekind's theorem to exhibit this concept clearly – \(\beta\), the point of the cut, identifies a unique separation between the two sets. Consequently, being familiar with Dedekind cuts is crucial for comprehending the structure of the real number line.