Math Problem Statement

What is the Completeness Property of Real Numbers ?

Solution

The Completeness Property of Real Numbers, also known as the Least Upper Bound Property or the Supremum Property, is a fundamental characteristic of the set of real numbers (R\mathbb{R}). It states:

Every non-empty subset of real numbers that is bounded above has a least upper bound (supremum) in the set of real numbers.

This means that for any set SRS \subseteq \mathbb{R}, if:

  1. SS is non-empty, and
  2. SS is bounded above (i.e., there exists some real number MM such that every element of SS is less than or equal to MM),

then there exists a real number LL (called the supremum or least upper bound) such that:

  • LL is an upper bound for SS (i.e., LsL \geq s for all sSs \in S), and
  • LL is the smallest such upper bound (i.e., if L<LL' < L, then LL' is not an upper bound of SS).

This property distinguishes the real numbers from other number sets, such as the rational numbers Q\mathbb{Q}, where certain bounded sets do not have a least upper bound within Q\mathbb{Q}.

Example:

Consider the set S={xQ:x2<2}S = \{ x \in \mathbb{Q} : x^2 < 2 \}, the set of all rational numbers whose square is less than 2. This set is bounded above (for example, by 2), but within the rational numbers, there is no least upper bound (since 2\sqrt{2} is irrational). However, in R\mathbb{R}, the supremum of this set is 2\sqrt{2}.

Would you like further details or have any questions?

Here are five related questions:

  1. What is the difference between the supremum and maximum of a set?
  2. How does the Completeness Property relate to the Dedekind Cut?
  3. Can the rational numbers (Q\mathbb{Q}) satisfy the Completeness Property?
  4. How does the Completeness Property connect with the Bolzano-Weierstrass Theorem?
  5. Why is the Completeness Property essential for calculus?

Tip: When working with bounded sets, always check if the set has a maximum element or only a supremum, as this distinction is important in analysis.

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Math Problem Analysis

Mathematical Concepts

Real Analysis
Set Theory
Bounded Sets

Formulas

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Theorems

Least Upper Bound Property
Supremum Property
Dedekind Cut

Suitable Grade Level

Undergraduate