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# Infinite sets and cardinalities

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### Infinite sets and cardinalities

1. 1. Infinite Sets andCardinalities {
2. 2. One-To-OneCorrespondence
3. 3. {3, 7, 11}{2, 4} { Example; non-equivalent sets
4. 4. If two non-empty sets havethe same cardinalnumber, they have a one-to-one correspondence { Cardinal number notation; n(A)
5. 5.  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 ratioCounting Numbers { 1, 2, 3, 4, 5, 6 ….. n }Whole Numbers; { 0, 1, 2, 3, 4, 5, …. n-1}
6. 6.  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
7. 7. This fact gives us a newdefinition for an infinite set;A set is infinite if it can be placed in aone-to-one correspondence with a propersubset of itself { Definition of an Infinite set
8. 8. • 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
9. 9. Countable Sets
10. 10. Sets that are not countable
11. 11.  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 numbersReal numbers; not countable
12. 12. Real numbers; not countable
13. 13. Infinite Set Cardinal NumberNatural/ Counting #sWhole NumbersIntegersRational NumbersIrrational NumbersReal Numbers { Summary