Cardinality and Bijections

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1 Countable and Cardinality and Bijections Gazihan Alankuş (Based on original slides by Brahim Hnich et al.) August 13, 2012

2 Countable and Countable and

3 Countable and How to count elements in a set? How many elements are in a set? Easy for finite sets, just count the elements What about infinite sets? Does it make sense to ask about the number of elements in an infinite set? Can we say this infinite set is larger than that infinite set? Which set is larger, the set of all integers or the set of all even integers?

4 Countable and Cardinality and Bijections If A and B are finite sets, it is not hard to see that they have the same cardinality if and only if there is a bijection from A to B Example: A = {1, 2, 3}, B = {a, b, c}, f = {(1, a), (2, b), (3, c)} Definition Sets A and B (finite or infinite) have the same cardinality if and only if there is a bijection from A to B

5 Countable and Cardinality and Bijections If A and B are finite sets, it is not hard to see that they have the same cardinality if and only if there is a bijection from A to B Example: A = {1, 2, 3}, B = {a, b, c}, f = {(1, a), (2, b), (3, c)} Definition Sets A and B (finite or infinite) have the same cardinality if and only if there is a bijection from A to B The set of all integers has the same cardinality as the set of even integers: f (x) = 2x is a bijection!!

6 Countable and Comparing cardinalities Let A and B be sets. We say A B if there is an injective function from A to B The function f (x) = x is an injective function from the set of even integers into the set of integers If there is an injective function from A to B, but not from B to A, we say A < B If there is an injective function from A to B and an injective function from B to A, we say A and B have the same cardinality

7 Countable and Example Let A be the closed interval [0, 1] (it includes the endpoints) and B be the open interval (0, 1) (it does not include the endpoints) There is a injective function f : A B such that f (x) = 1 3 x There is a injective function g : B A such that g(x) = x Therefore A and B have the same cardinality

8 Countable and Countable and Uncountable Definition A set A is countable if A N. This is because an injective function from A to N can be viewed as assigning numbers to the elements of A, thus counting them Sets that are not countable are uncountable

9 Countable and Examples of Countable Sets The set of all integers: 0, 1, 1, 2, 2, 3, 3,... The set of odd numbers The set of positive rational numbers The cardinality of the set of natural numbers is denoted by N 0

10 Countable and The Smallest Infinite Set Theorem If A is an infinite set, then A > N 0

11 Countable and Can we make a list of real numbers? Every real number can be represented as an infinite decimal fraction like: Suppose we have constructed a list of all real numbers

12 Countable and 1. a 10.a 11 a 12 a 13 a 14 a 15 a a 20.a 21 a 22 a 23 a 24 a 25 a a 30.a 31 a 32 a 33 a 34 a 35 a a 40.a 41 a 42 a 43 a 44 a 45 a a 50.a 51 a 52 a 53 a 54 a 55 a We construct the following number 0.b 1 b 2 b 3 b 4 b 5 b 6... such that b i a ii for all i.

13 Countable and 1. a 10.a 11 a 12 a 13 a 14 a 15 a a 20.a 21 a 22 a 23 a 24 a 25 a a 30.a 31 a 32 a 33 a 34 a 35 a a 40.a 41 a 42 a 43 a 44 a 45 a a 50.a 51 a 52 a 53 a 54 a 55 a Thus 0.b 1 b 2 b 3 b 4 b 5 b 6... is not in the list!

Sets. Gazihan Alankuş (Based on original slides by Brahim Hnich et al.) August 6, Outline Sets Equality Subset Empty Set Cardinality Power Set

Sets. Gazihan Alankuş (Based on original slides by Brahim Hnich et al.) August 6, Outline Sets Equality Subset Empty Set Cardinality Power Set Gazihan Alankuş (Based on original slides by Brahim Hnich et al.) August 6, 2012 Gazihan Alankuş (Based on original slides by Brahim Hnich et al.) Gazihan Alankuş (Based on original slides by Brahim Hnich