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CHAPTER 2

CHAPTER 2. Set Theory. 2.2. Subsets. Objectives Recognize subsets and use the notation  . Recognize proper subsets and use the notation . Determine the number of subsets of a set. Apply concepts of subsets and equivalent sets to infinite sets. 3. Subsets.

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CHAPTER 2

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  1. CHAPTER 2 Set Theory

  2. 2.2 Subsets

  3. Objectives Recognize subsets and use the notation . Recognize proper subsets and use the notation . Determine the number of subsets of a set. Apply concepts of subsets and equivalent sets to infinite sets. 3

  4. Subsets • The notation AB means that Ais not a subset of B. Set A is not a subset of set B if there is at least one element of set A that is not an element of set B. • Every set is a subset of itself. 4

  5. Example 1: Using the Symbols  and  Write  or  in the blank to form a true statement. A = {1, 3, 5, 7} B = {1, 3, 5, 7, 9, 11 } A ____ B Set A is a subset of B, A B A = {x│x is a letter in the word proof} A = {y│y is a letter in the word roof} A ____ B Set A is not a subset of B, A B 5

  6. Proper Subsets 6

  7. Example 2: Proper Subsets Write ,  , or both in the blank to form a true statement: A = {x | x is a person and x lives in San Francisco} B = {x | x is a person and x lives in California} A ____B Solution: A , B • A = { 2, 4, 6, 8} B = { 2, 8, 4, 6} A ____B Solution: A B 7

  8. Subsets and the Empty Set 8

  9. The Number of Subsets of a Given Set • As we increase the number of elements in the set by one, the number of subsets doubles. • The number of subsets of a set with n elements is 2n. • The number of proper subsets of a set with n elements is 2n – 1. 9

  10. Example 4: Finding the Number of Subsets and Proper Subsets Find the number of subsets and the number of proper subsets. • {a, b, c, d, e } There are 5 elements so there are 25 = 32subsets and 25 − 1 = 31 proper subsets. b. {x | x  and 9 ≤x≤ 15 } In roster form,we see that there are 7 elements: { 9, 10, 11, 12, 13, 14, 15 } There are 27 = 128subsets and 27 −1 = 127 proper subsets. 10

  11. The Number of Subsets of Infinite Sets • There are א0 natural numbers. • It has 2א0 subsets. • It has 2א0 – 1 proper subsets • 2א0 > א0 • Denote 2א0 by א1 • א1 > א0 • א0 is the “smallest” transfinite cardinal number in an infinite hierarchy of different infinities. 11

  12. Cardinal Numbers of Infinite Sets Georg Cantor (1845 – 1918) studied the mathematics of infinity and assigned the transfinite cardinal numberא0 to the set of natural numbers. He used one-to-one correspondences to establish some surprising equivalences between the set of natural numbers and its proper subsets. 12

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