INFIINITE SETS

1. INFIINITE SETS CSC 172 SPRING 2002 LECTURE 23

2. Cardinality and Counting The cardinality of a set is the number of elements in the set Two sets are equipotent if and oly if they have the same cardinality The existence of a one-to-one correspondence between two sets proves that they are equipotent Counting is just creating a one-to-one correspondence between set and the set of integers from 1 to some number n

3. Example

4. Finite & Infinite Sets Can you create a one-to-one correspondence between a set and a proper subset of itself? If you can, you have a solution to x = x + y, where x is the cardinality of the set and y >= 1 is the cardinality of the stuff you removed Impossible with finite sets Possible with infinite sets Technically, an infinite set is a set where there exists a one-to-one correspondence between the set itself and a proper subset

5. Example Let N be the set of integers > 0 {1,2,3,…} Clearly N –{1} is a proper subset of N {2,3,4,….} We can create a 1:1 correspondence between the two sets by matching each element x N with element x+1 N - {1} Therefore, N is an infinite set.

6. Countable Infinity Once we have an infinite set, we can prove another set infinite by creating a one-to-one correspondence between the known-infinite set and a subset (possibly the whole set) of theother set For example, The set of all integers Z {.., -2,-1,0,1,2,…} contains N, (N is 1:1 with itself) so Z is infinite.

7. … 4, 3, 2, {1 {…,-2,-1,0,1,2,..} Z and N Z and N are actually equipotent. What is the 1:1? {1,2,3,4,…} {…-3,-2,-1,0,1,2,3,…} Each element x N maps To (x/2) if x is odd Or –(x/2) if x is even The mapping goes both ways, so Z is equipotent with N

8. Pairs to N

9. Another mapping The set N is equipotent with the set of pairs of positive integers Therefore, the set of rational numbers Q is also equipotent with Z and N, since very rational number can be represented by a pair of integers

10. 0 Many common infinite sets are equipotent with the set of integers This cardinality is written 0 Pronounced “aleph-naught” A set with this cardinality is said to be countably infinite because we can put its elements in a 1:1 correspondence with N

11. Uncountable Infinity The set R of real numbers is infinite since it contains all the integers. Is it countably infinite?

12. 1 r=1/ 0 R is equipotent with the set of reals between 0 and 1 x x=arctan(x)+( /2)+ ) y=tan(y-(/2)) y

13. Disproving Equipotency If you can find a 1:1 correspondence, you have proven equipotency If you cannot find a 1:1 correspondence you have proven nothing, either way. To disprove equipotency you must prove that no 1:1 correspondence is possible Alternately, if you prove one infinity countable and another infinity uncountable you have proven that the two sets are not equipotent

14. There is no one-to-one correspondence between the reals (0,1) and N Suppose there was (some table listing all the real numbers) We can construct a new number not on the table The value of the ith digit depends on the ith digit of the ith real in the table If said digit is 0-4, new digit is “8” If said digit is 5-9, new digit is “1”

15. Diagonalization If the new real was all ready on the table (the rth), then it would have contributed the rth digit to the new number But the rth digit of the new number differs by at least 4 digits