Interactive Real Analysis - part of MathCS.org

• Real Analysis
• 1. Sets and Relations
• 2. Infinity and Induction
• 2.1. Countable Infinity
• 2.2. Uncountable Infinity
• 2.3. The Principle of Induction
• 2.4. The Real Number System
• 2.5. Pi and e are irrational
• 3. Sequences of Numbers
• 4. Series of Numbers
• 5. Topology
• 6. Limits, Continuity, and Differentiation
• 7. The Integral
• 8. Sequences of Functions
• 9. Historical Tidbits
• Java Tools

Examples 2.3.2(b):

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Which of the following sets are well-ordered ?

1. The number systems N, Z, Q, or R ?
2. The set of all rational numbers in [0, 1] ?
3. The set of positive rational numbers whose denominator equals 3 ?

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The natural numbers N are well-ordered:

A subset of natural numbers may not have a largest element, but it must have a smallest element.

The integers Z are not well-ordered:

While many subsets of Z has a smallest element, the set Z itself does not have a smallest element.

The rationals Q are not well-ordered:

The set Q itself does not have a smallest element.

The real numbers R are not well-ordered:

R itself does not have a smallest element.

The set of all rational numbers in [0, 1] is not well-ordered:

While the set itself does have a smallest element (namely 0), the subset of all rational numbers in (0, 1) does not have a smallest element.

The set of all positive rational numbers whose denominator equals 3 is well-ordered:

This set is actually the same as the set of natural numbers, because we could simply re-label a natural number n to look like the symbol n / 3. Then both sets are the same, and hence this set is well-ordered.