Aim: How can the word ‘infinite’ define a collection of elements? Do Now: What is a googolplex? In the PBS science program Cosmos: A Personal Voyage, Episode 9: "The Lives of the Stars", astronomer and television personality Carl Sagan estimated that writing a googolplex in numerals (i.e., "10,000,000,000...") would be physically impossible, since doing so would require more space than the known universe provides.

One-to-one Correspondence Number and its place on a number line -7.5 -1 2 6 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 Mapping relations -3 -1 1 3 5 relation A 2 4 A relation A–1 A–1 1 2 3 4 -3 -1 5

One-to-One Function Only one-to-one functions have inverses that are functions. A one-to-one function passes the horizontal line test i.e. the line crosses the function at one and only one point. A function f has an inverse function f-1 if and only if f is one-to-one.

One-to-One Correspondence A one-to-one correspondence (1 – 1 correspondence) between two sets A and B is a rule or procedure that pairs each element of A with exactly one element of B and each element of B with exactly one element of A. Consider: Concert hall has 890 seats For one performance all seats are occupied no need to count to know attendance For another performance six seats are empty. no need to count to know attendance: 890 – 6 = 884.

One-to-One Correspondence Equivalent Sets – contain the same number of elements. Cardinalities are equal: n(A) = n(B) or |A| = |B|. expanded Two sets A and B are equivalent, denoted by A ~ B, if and only if A and B can be placed in a one-to-one correspondence. A = {a, b, c, d, e} n(A) = n(B) |A| = |B| = 5 B = {1, 2, 3, 4, 5} A ~ B A = {a, b, c, d, e} B = {1, 2, 3, 4, 5}

Model Problem Establish a one-to-one correspondence between the set of natural numbers N = {1, 2, 3, 4, 5, . . . , n, . . .} and the set of even natural numbers E = {2, 4, 6, 8, 10, . . . , 2n, . . .} N = {1, 2, 3, 4, 5, . . . , n, . . .} n  N E = {2, 4, 6, 8, 10, . . . , 2n, . . .} 2n  N determines exactly what elements are paired in N & E n  2n 17  34 76867  153734 and establishes a one-to-one correspondence between sets N ~ E

Definition of Infinite Set A set is an infinite set if it can be placed in a one-to-one correspondence with a proper subset of itself. N ~ E Can the set {1, 2, 3} be placed in a one-to- one correspondence with one of its proper subsets? No, {1, 2, 3} is finite and every proper subset of {1, 2, 3} has 2 or fewer elements: {1}; {1, 2}; {2, 3}, etc.

Model Problem Verify the S = {5, 10, 15, 20, . . . 5n, . . .} is an infinite set. plan: establish a one-to-one correspondence with a proper subset S = {5, 10, 15, 20, . . . 5n, . . .} 5n  S T = {10, 20, 30, 40, . . . 10n, . . .} 10n  T This general correspondence 5n  10n establishes a one-to-one with one of S’s proper subsets (T) making S an infinite set.

Model Problem Verify the V = {40, 41, 42, 43, . . . 40 + n, . . .} is an infinite set. plan: establish a one-to-one correspondence with a proper subset

Cardinality of Infinite Sets Does an infinite set have cardinality? Georg Cantor ‘How can one infinity be greater than another?” o (read as aleph-null) represents the cardinal number for the set N of natural numbers. n(N) = o Since o has a cardinality greater than any infinite number it is called a transfinite number. Many infinite sets have a cardinality of o.

The set of integers has a cardinality of o. Model Problem Show that the set of integers I = {. . . , -4, -3, -2, -1, 0 , 1, 2, 3, 4, . . .} has a cardinality of o. The set of integers has a cardinality of o. I = {. . . , -4, -3, -2, -1, 0 , 1, 2, 3, 4, . . .} N = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10. 11, . . . } no obvious one-to-one correspondence however, with rearranging of I N = {1, 2, 3, 4, 5, 6, 7, 8, . . . , 2n – 1, 2n, . . . } I = {0, 1, -1, 2, -2, 3, -3, 4, . . . , –n + 1, n, . . .} each even natural number 2n of set N is paired with the integer n of set I. (blue arrows) each odd natural number 2n – 1 of N is paired with the integer –n + 1 of I. (red arrows) general correspondence (2n)  n and (2n – 1)  (-n + 1): 1-2-1

Model Problem

Theorem of Rational Numbers The set Q+ of positive rational numbers is equivalent to the set N of natural numbers. expressed in lowest terms; smallest to largest Which rational is assigned to N = 11? Rational number in array Corresponding natural number 1 2 3 4 5 6 7 8 9 10

Countable Set A set is a countable set if and only if it is a finite set or an infinite set that is equivalent to the set of natural numbers. Every infinite set that is countable has a cardinality of o. However, not all infinite sets are countable. Ex: Theorem: The set of A = {x|x  R and 0 < x < 1} is not a countable set. Proof by contradiction – Assume it is a countable set and work until we arrive at a contradiction.

Proof by Contradiction The set of A = {x|x  R and 0 < x < 1} is not a countable set. assume countable N = {1, 2, 3, 4, 5, 6, 7, 8, . . . n, . . . } A = {a1, a2, a3, a4, a5, a6, a7, a8, . . . an, . . . } build a number d using diagonal method 1  a1 = 0.3573485 . . . 2  a2 = 0.0652891 . . . 3  a3 = 0.6823514 . . . 0 < < 1 0.4 7 3 1 . . . 4  a4 = 0.0500310 . . . 0 < d < 1 n  an = 0.3155728 . . . 5 . . . contradiction!!! but it’s not is set A! It differs from each number of A in at least 1 decimal place

Countable/Uncountable An infinite set that is not countable is said to be uncountable. An uncountable set does not have a cardinality of o. It has a cardinality of c, from the word continuum. c > o Cardinality of Some Infinite Sets Set Cardinal Number Natural Numbers, N o Integers, I Rational Numbers, Q Irrational Numbers c Any set of form {x|a < x < b} a and b are real a  b Real Numbers, R

This set is called the Power Set of S denoted by P(S) Cantor’s Theorem Cantor’s Theorem Let S be any set. The set of all subsets of S has a cardinal number that is larger that the cardinal number of S. This set is called the Power Set of S denoted by P(S) recall: S = {1, 2, 3} |S| = 3 S has 23 = 8 subsets Cantor said this is true for infinite sets as well No matter how large the cardinal number of a set we can find a set that has a larger cardinal number There are infinitely many transfinite numbers

Transfinite Arithmetic Theorems For any whole number a, o + a = o and o – a = o o + o = o and in general, o + o + o + o = o a finite number of o c + c = c and in general, c + c + c + c = c a finite number of c c + o = c c o = c