Cardinality of sets. Cardinality of sets

Transcription

1 Cardinality of sets Two sets A and B have the same size, or cardinality, if and only if there is a bijection f : A Ñ B. Example: We know that set ta, b, cu has elements because we can count them: 1: a : b : c But this is essentially the same as the bijection f 1 c b a A B Cardinality of sets Definition: Two sets A and B have the same size, orsame cardinality, ifand only if there is a bijection f : A Ñ B. (This allows us to measure the relative sizes of sets, even if they happen to be infinite!) Example: The sets Z 0 and Z 0 have the same cardinality since f : Z 0 Ñ Z 0 x fiñ x ` 1 is a bijective map. (A function is bijective if and only if it is invertible It is invertible, with inverse x fiñ x 1.)

2 Countably infinite sets A set is countable if it is either finite or the same cardinality as the natural numbers (N Z 0 ). If a set A is not finite but is countable, we say A is countably infinite and write 0 (pronounced aleph naught or aleph null ). To show that 0 :showa is not finite, and give a bijection f : Z 0 Ñ A. Examples: 1. Z 0 is countably infinite: It is not finite, and f : Z 0 Ñ Z 0 by x fiñ x is a bijection.. Z 0 is countably infinite: It is not finite, and f : Z 0 Ñ Z 0 by x fiñ x ` 1 is a bijection.. Z is countably infinite: Not finite, and f : Z 0 Ñ Z 0 by x fiñ p 1q x rx{s is a bijection. Z 0 : Z: More on this last example, 0 : We started with the picture Z 0 : Z: This at least gives us a list of integers, 0:0, 1: 1, :1, :, 4:,... If I know that every integer appears on this list somewhere, then I know that the integers are countable. (Ok answer) The next step in giving a more sophisticated, more robust, answer is to try to get the formula # written down: x{ if x is even, f : Z 0 Ñ Z x fiñ (Better answer) px ` 1q{ if x is odd. To be even more sophisticated, we used the ceiling function to get a closed form# answer: x{ if x is even, px ` 1q{ if x is odd p 1qx rx{s, for x P Z 0.(Best answer)

3 Recall that A B if and only if there is a bijection f : A Ñ B. If we know that 0 and f : A Ñ B is a bijection, then B 0. Example: To show that Z teven integers u is countably infinite, we could construct a bijection like in the previous example. But it s a little more straightforward to note that f : Z Ñ Z is a bijection, x fiñ x so that Z 0. You try: For each of the following, show that the set is countably infinite. (Define a bijective function to something that we know to be countably infinite if it s not too hard; otherwise, explain how to make the list.) 1. The set of negative integers.. The set of integers less than The set of integers that are integer multiples of. 4. The set of integers that are not integer multiples of.

4 The rational numbers Claim: Q 0 is countably infinite. Start with 0 fiñ 0. Thenmakea table: {1 {1 5 {1 6 4{1 11 5{1 1{ { 7 { 4{ 5{ 4 1{ 8 { { 4{ 5{ (skip prev. counted fractions) 4 9 1{4 {4 {4 4{4 5{ {5 {5 {5 4{5 5{5 Then use the same alternating map that we did for Z 0 Ñ Z to build a bijection Q 0 Ñ Q. Are there sets that are not countable? Claim: The set of real numbers in the interval r0, 1q is not countable. Proof technique: Proof by contradiction. To prove that blah is not true: Suppose blah is true, and find a logical contradiction. So to prove our claim, we suppose that the set r0, 1q is countable. This would mean that the real number in r0, 1q can be listed. Take one such list. Goal: To get the contradiction, we ll show that the list isn t actually complete! Namely, algorithmically produce an element of r0, 1q that isn t on the list.

5 Take this is the supposedly complete list of real numbers in r0, 1q. For example: Algorithm for producing a number that is not on the list: In the ith number in the list, highlight the i 1st digit. Build a new number as follows: i. If the highlighted digit of the ith number is a 0, then make the corresponding digit of the new number a ii. If the highlighted digit of the ith number is not a 0, then make the corresponding digit of the new number a Example: In this way, this new number di ers from every item in the list in at least one digit! Are there sets that are not countable? Claim: The set of real numbers in the interval r0, 1q is not countable. Proof technique: Proof by contradiction. To prove that blah is not true: Suppose blah is true, and find a logical contradiction. So to prove our claim, we suppose that the set r0, 1q is countable. This would mean that the real number in r0, 1q can be listed. Using our algorithm, any such list will be missing at least one number. Thus we arrive at a contradiction. Therefore, our assumption was false, and so the set r0, 1q is not countable.

6 More facts Unions If A and B are countable, then so is A Y B. Therefore, if A 1,A,...,A n are all countable, then so is î n i 1 A i. In fact, if ta x x P Cu is a countable collection of countable sets (i.e. C is a countable set), then so is î xpc A x. Example: Let A x ty{x y P Zu for each x P Z 0.Then A x Q, which is countable. xpz 0 However, if ta x x P Uu is an uncountable collection of countable sets, then î xpu A x could be countable or uncountable (we can t tell). Countable: î xpr0,1q A x where A x Q. Uncountable: î xpr0,1q A x where A x txu. Containment If A is not countable and A Ñ B, thenb is not countable. Ex: Any subset of Z is countable. If B is countable and A Ñ B, thena is countable (prove on HW). Ex. Since r0, 1q is not countable, then neither is R.

7 Comparing sizes If there is an injective function f : A Ñ B then we write A B. (Makes sense for finite sets; take as a definition for infinite sets.) Theorem (Schröder-Bernstein Theorem) If A B and B A, then A B. Example We can show p0, 1q p0, 1s by using the fact that f : p0, 1q Ñp0, 1s x fiñ x and g : p0, 1s Ñp0, 1q x fiñ x{ are both injective, even though they are not surjective nor are they inverses of each other.