Infinite sets and cardinalities
- 3. {3, 7, 11} {2, 4} { Example; non-equivalent sets
- 4. If two non-empty sets have the same cardinal number, they have a one-to- one correspondence { Cardinal number notation; n(A)
- 6. Intuitively, this seems incorrect. Counting numbers should have one less element than whole numbers since they start at 0 instead of 1, right? (Galileo’s Paradox) Since they are infinite, however, we have a one- to-one ratio Counting Numbers { 1, 2, 3, 4, 5, 6 ….. n } Whole Numbers; { 0, 1, 2, 3, 4, 5, …. n-1}
- 7. A Proper Subset of a set has least one less element than that set P= {2, 3, 6, 9} A PROPER subset would be {3, 6, 9} Counting Numbers are a proper Subset of Whole numbers (counting numbers are all the same numbers, excluding 0) Back to Proper Subsets
- 8. This fact gives us a new definition for an infinite set; A set is infinite if it can be placed in a one-to-one correspondence with a proper subset of itself { Definition of an Infinite set
- 9. • The set of Integers; {…, -4, -3, -2, -1, 0, 1, 2, 3, 4, …} How can we show a one-to-one correspondence? {1, 2, 3, 4, 5, 6, 7, …} Subset of the set of integers {0, 1, -1, 2, -2, 3, -3, ….} { Example; Show the set of integers is an infinite set
- 10. Countable Sets
- 11. Sets that are not countable
- 12. The set of real numbers are all numbers that can be written as decimals. Because there is an infinite continuum from, say, 1 to 2, you cannot set up a one-to-one correspondence 1, 1.1, 1.01, 1.001…. 1.12, 1.13, 1.14 You can keep adding more and more numbers between 1 & 2 In between every number, there is an infinite amount of numbers Real numbers; not countable
- 13. Real numbers; not countable
- 14. Infinite Set Cardinal Number Natural/ Counting #s Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers { Summary