INFIINITE SETS CSC 172 SPRING 2002 LECTURE 23

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

Example  1  2  3  4  5

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

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.

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.

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 … 4, 3, 2, {1 {…,-2,-1,0,1,2,..}

Pairs to N … (1,5)(2,5)(3,5)(4,5) … (1,4)(2,4)(3,4)(4,4) (1,3)(2,3)(3,3)(4,3) (1,2)(2,2)(3,2)(4,2) (1,1)(2,1)(3,1)(4,1)

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

0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

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

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

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

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”

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