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Chapter 1: Problem 21

The rational numbers \(\mathbb{Q}\) satisfy the axioms for an ordered field. Show that the completeness axiom would not be satisfied. That is show that this statement is false: Every nonempty set \(E\) of rational numbers that is bounded above has a least upper bound (i.e., \(\sup E\) exists and is a rational number).

Short Answer

Expert verified
The Completeness Axiom is false in \(\mathbb{Q}\), as some sets like \(\{x \in \mathbb{Q} \,|\, x^2 < 2\}\) have no rational supremum.

Step by step solution

01

Understand the Completeness Axiom

The Completeness Axiom states that every nonempty set of real numbers that is bounded above has a least upper bound (supremum) in the real numbers. The goal is to show that this axiom is not satisfied when considering rational numbers instead of real numbers.
02

Identify a Suitable Set

Choose a nonempty bounded above set of rational numbers that lacks a rational least upper bound. A common choice for this demonstration is the set of rational numbers whose square is less than 2, i.e., \(E = \{ x \in \mathbb{Q} \,|\, x^2 < 2 \}\).
03

Show Boundedness of the Set

Demonstrate that the set \(E\) is bounded above. The real number \(\sqrt{2}\) is not rational, but if \(x \in E\), then \(x^2 < 2\). Furthermore, \(x \leq 2\) can serve as an upper bound for \(E\) since if \(x > 2\), \(x^2 > 4\), which is greater than 2.
04

Prove No Rational Supremum Exists

Assume there exists a rational number \(s\) that is the least upper bound of \(E\). Then \(s^2\) must equal 2 for \(s\) to be the smallest possible upper bound. However, no rational number squared equals 2, which contradicts the assumption that such an \(s\) is rational, hence proving no rational \(\sup E\) exists.
05

Conclusion

We have shown that there is no rational number that can act as the least upper bound for the set \(E\). This implies that the set \(E\) is bounded above but does not satisfy the Completeness Axiom in the rational numbers, demonstrating that \(\mathbb{Q}\) is not complete.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Ordered Field
An ordered field is a field that is equipped with an ordering. This means we can compare any two elements to determine which is greater or smaller. In an ordered field, the order satisfies certain properties:
  • Transitivity: If \(a < b\) and \(b < c\), then \(a < c\).
  • Trichotomy: For any two elements \(a\) and \(b\), one and only one is true: \(a < b\), \(a = b\), or \(a > b\).
  • Additive property: If \(a < b\), then \(a + c < b + c\) for any \(c\).
  • Multiplicative property: If \(a < b\) and \(c > 0\), then \(ac < bc\).
The rational numbers \(\mathbb{Q}\) fit these criteria, classifying them as an ordered field. You can naturally arrange them on a number line and perform arithmetic operations that preserve the order.
Completeness Axiom
The completeness axiom is a central concept in real analysis. It states that any nonempty set of real numbers, which has an upper bound, must have a least upper bound (also known as a supremum) that is also a real number. However, this axiom does not apply to the rational numbers. The exercise demonstrates this by showing that for the set \( E = \{ x \in \mathbb{Q} \,|\, x^2 < 2 \} \), there is no rational number that serves as the least upper bound. This failure to satisfy the completeness axiom indicates that the rational numbers are not a complete field.
Least Upper Bound
The least upper bound, or supremum, of a set is the smallest number that is greater than or equal to every number in the set. To illustrate: for the set \(E = \{ x \in \mathbb{Q} \,|\, x^2 < 2 \} \), any real number greater than \(\sqrt{2}\) would be an upper bound, but \(\sqrt{2}\) itself is the least upper bound in the real numbers. However, since \(\sqrt{2}\) is not a rational number, the set \(E\) in the rational number system lacks a rational least upper bound. This is a key point in understanding why \(\mathbb{Q}\) is not complete; it can't include all limits of all convergent sequences of rationals.
Bounded Set
A set is termed as bounded if all its members are confined within two fixed bounds. Specifically, a set \(E\) is bounded above if there is a number greater than or equal to every element in \(E\). For example, consider \(E = \{ x \in \mathbb{Q} \,|\, x^2 < 2 \} \). The set \(E\) is bounded above by any number greater than \(\sqrt{2}\). This indicates the presence of an upper bound, but the set lacks a rational number that is the least upper bound. Understanding bounded sets helps in grasping the broader concept of completeness and where \(\mathbb{Q}\) falls short as compared to \(\mathbb{R}\).

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Most popular questions from this chapter

Use this definition of "dense in a set" to answer the following questions: A set \(E\) of real numbers is said to be dense in a set \(A\) if every interval \((a, b)\) that contains a point of \(A\) also contains a point of \(E .\) (a) Show that dense in the set of all reals is the same as dense. (b) Give an example of a set \(E\) dense in \(\mathbb{N}\) but with \(E \cap \mathbb{N}=\emptyset\). (c) Show that the irrationals are dense in the rationals. (A real number is irrational if it is not rational, that is if it belongs to \(\mathbb{R}\) but not to \(\mathbb{Q} .)\) (d) Show that the rationals are dense in the irrationals. (e) What property does a set \(E\) have that is equivalent to the assertion that \(\mathbb{R} \backslash E\) is dense in \(E ?\)

$$ \text { Show that }|x|=\max \\{x,-x\\} \text { . } $$

Let \(S\) be a set consisting of two elements labeled as \(A\) and \(B\). Define \(A+A=\) \(A, B+B=A, A+B=B+A=B, A \cdot A=A, A \cdot B=B \cdot A=A\), and \(B \cdot B=B\). Show that all nine of the axioms of a field hold for this structure.

$$ \text { Show for every nonempty, finite set } E \text { that } \sup E=\max E . $$

Find \(\sup E\) and inf \(E\) and (where possible) \(\max E\) and \(\min E\) for the following examples of sets: (a) \(E=\mathbb{N}\) (b) \(E=\mathbb{Z}\) (c) \(E=\mathbb{Q}\) (d) \(E=\mathbb{R}\) (e) \(E=\\{-3,2,5,7\\}\) (f) \(E=\left\\{x: x^{2}<2\right\\}\) (g) \(E=\left\\{x: x^{2}-x-1<0\right\\}\) (h) \(E=\\{1 / n: n \in \mathbb{N}\\}\) (i) \(E=\\{\sqrt[n]{n}: n \in \mathbb{N}\\}\)
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