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PowerPoint Presentation . Section 2.6 Infinite Sets. The idea of Infinity . Infinity is not a number. Infinity is an idea related to numbers. It is believed that there are an infinite number of stars in the universe -- the number of stars cannot be counted. Infinite Sets.

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  1. PowerPoint Presentation Section 2.6 Infinite Sets

  2. The idea of Infinity • Infinity is not a number. • Infinity is an idea related to numbers. • It is believed that there are an infinite number of stars in the universe -- the number of stars cannot be counted.

  3. Infinite Sets • An infinite set contains an uncountable number of elements. • One way to show that a set is infinite is by using the definition of infinite sets: An infinite set can be placed in a one-to-one correspondence with a proper subset of itself

  4. The Set of Natural Numbers is an Infinite Set … • The Natural Numbers is the set: { 1,2,3,4,…} • A Proper Subset of the Natural Numbers is the set: { 2,3,4,5…} • These sets can be put in a one-to-one correspondence.

  5. Input-Output Table for One-to-One Correspondence

  6. Input-Output Table for One-to-One Correspondence

  7. General Term for One-to-One Correspondence

  8. The Even Numbers are an Infinite Set … …

  9. Input-Output Table for One-to-One Correspondence

  10. Finding the General Terms for One-to-One Correspondence

  11. General Terms for One-to-One Correspondence

  12. The Multiples of 10 are an infinite set • We can use any proper subset of the infinite set for the one-to-one correspondence. We pick one with a pattern so we can name the general term: • 10, 20, 30, 40… • 20,40,60,80… • 100, 200, 300…

  13. Input-Output Table for One-to-One Correspondence

  14. Finding the General Terms for One-to-One Correspondence

  15. Finding the General Terms for One-to-One Correspondence

  16. Countable Sets and Aleph-Null • An infinite set is countable if it can be put in a one-to-one correspondence with the Natural Numbers. • Countable sets have a cardinal number of “aleph-null”. The symbol for aleph-null is:

  17. Different types of Infinity

  18. Arithmetic of Aleph-Null + = - 10 =

  19. Arithmetic of Aleph-Null 3 + = ? 5 - 4 = ?

  20. Arithmetic of Aleph-Null 3 + = 5 - 4 = 0

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