中山大学 现代数学引论

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Sets

Sets .................................................................................................................................................................. 1

1. Operations with Sets................................................................................................................................ 2

1.1. Basic Operations .............................................................................................................................. 2

1.2. The Limit of a Sequence of Sets ...................................................................................................... 3

3.2. Equivalent Relations & Classification............................................................................................ 16

3.3. Ordering Relations & Zorn’s Lemma............................................................................................. 18

4. The Completeness of Real Numbers ..................................................................................................... 21

Problems .................................................................................................................................................... 27

Appendix: Cantor’s Ternary Set................................................................................................................ 29

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1. Operations with Sets

1.1. Basic Operations

Definition Let

Definition Let A and B be two sets, then the difference of A and B is defined as

Theorem (De. Morgan Law) Let A be a set and

be a family of sets, then

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1.2. The Limit of a Sequence of Sets

Definition Let

,

,

is then called a monotone-decreasing sequence of sets.

Example Let

,

,

is then a monotone-increasing sequence of sets.

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Example Let

Proof:

, there must be a positive integer

In this way, a subsequence

of