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(a):

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Which of the following sets and their ordering relations are partially ordered, ordered, or well-ordered:

1. S is any set. Define a b if a = b
2. S is any set, and P(S) the power set of S. Define A B if A B
3. S is the set of real numbers between [0, 1]. Define a b if a is less than or equal to b (i.e. the 'usual' interpretation of the symbol )
4. S is the set of real numbers between [0, 1]. Define a b if a is greater than or equal to b.

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1. S is any set. Define a b if a = b.

This is a trivial partial ordering. Since no element is related to an element different from itself, this is not an ordered set. Without more information about S we can not determine anything else.

This example shows that any set can be partially ordered.

2. S is any set, and P(S) the power set of S. Define A B if A B

Recall that if A B and B A then A = B. Therefore, this is indeed a partial ordering. Without more information about the set S we can not determine anything else.

3. S is the set of real numbers between [0, 1]. Define a b if a is less than or equal to b (i.e. the 'usual' interpretation of the symbol )

This is clearly an ordering, and the set [0, 1] with this ordering is usually represented as a subset of the number line. It is not a well-ordered set, because the subset (0,1] has no smallest element.

4. S is the set of real numbers between [0, 1]. Define a b if a is greater than or equal to b.

This is also an ordering. The set is not well-ordered, however, because the subset [0, 1) has no smallest element. Note that 1 is the smallest element in the set [0, 1), according to our convention. Here it is important to distinguish between the conventional meaning of the symbol and its meaning as we choose to define it for a particular situation.