Countability Jason Filippou CMSC250 @ UMCP 06-23-2016 Jason Filippou (CMSC250 @ UMCP) Countability 06-23-2016 1 / 12
Outline 1 Infinity 2 Countability of integers and rationals 3 Uncountability of R Jason Filippou (CMSC250 @ UMCP) Countability 06-23-2016 2 / 12
Infinity Infinity Jason Filippou (CMSC250 @ UMCP) Countability 06-23-2016 3 / 12
Infinite sets Infinity Definition (Finite set) Let n N. A set A is called finite if and only if: 1 A =, or 2 There exists a bijection from the set {1, 2,..., n} to A. Definition (Infinite set) A set A is called infinite if, and only if, it is not finite. Jason Filippou (CMSC250 @ UMCP) Countability 06-23-2016 4 / 12
Countable sets Infinity Definition (Countable set) Let A be any set. A is countable if, and only if: 1 A is finite, or 2 There exists a bijection from N to A. Jason Filippou (CMSC250 @ UMCP) Countability 06-23-2016 5 / 12
Countable sets Infinity Definition (Countable set) Let A be any set. A is countable if, and only if: 1 A is finite, or 2 There exists a bijection from N to A. In the second case, A can also be called countably infinite. Jason Filippou (CMSC250 @ UMCP) Countability 06-23-2016 5 / 12
Countable sets Infinity Definition (Countable set) Let A be any set. A is countable if, and only if: 1 A is finite, or 2 There exists a bijection from N to A. In the second case, A can also be called countably infinite. Definition (Uncountable set) A set A is called uncountable if, and only if, it is not countable. Jason Filippou (CMSC250 @ UMCP) Countability 06-23-2016 5 / 12
Countability of integers and rationals Countability of integers and rationals Jason Filippou (CMSC250 @ UMCP) Countability 06-23-2016 6 / 12
Z is countable Countability of integers and rationals... 3 2 1 0 1 2 3... Jason Filippou (CMSC250 @ UMCP) Countability 06-23-2016 7 / 12
Z is countable Countability of integers and rationals... 3 2 1 0 1 2 3...... 7 5 3 1 2 4 6... Jason Filippou (CMSC250 @ UMCP) Countability 06-23-2016 7 / 12
Z is countable Countability of integers and rationals... 3 2 1 0 1 2 3...... 7 5 3 1 2 4 6... Let s call this function f. We can make the following observations about f: Jason Filippou (CMSC250 @ UMCP) Countability 06-23-2016 7 / 12
Z is countable Countability of integers and rationals... 3 2 1 0 1 2 3...... 7 5 3 1 2 4 6... Let s call this function f. We can make the following observations about f: 1 No integer is counted twice! So, f is...? Jason Filippou (CMSC250 @ UMCP) Countability 06-23-2016 7 / 12
Z is countable Countability of integers and rationals... 3 2 1 0 1 2 3...... 7 5 3 1 2 4 6... Let s call this function f. We can make the following observations about f: 1 No integer is counted twice! So, f is...? 1-1. 2 All integers are (eventually) accounted for! So, f is...? ason Filippou (CMSC250 @ UMCP) Countability 06-23-2016 7 / 12
Z is countable Countability of integers and rationals... 3 2 1 0 1 2 3...... 7 5 3 1 2 4 6... Let s call this function f. We can make the following observations about f: 1 No integer is counted twice! So, f is...? 1-1. 2 All integers are (eventually) accounted for! So, f is...? onto. ason Filippou (CMSC250 @ UMCP) Countability 06-23-2016 7 / 12
Z is countable Countability of integers and rationals... 3 2 1 0 1 2 3...... 7 5 3 1 2 4 6... Let s call this function f. We can make the following observations about f: 1 No integer is counted twice! So, f is...? 1-1. 2 All integers are (eventually) accounted for! So, f is...? onto. From (1) and (2) we can deduce that the function is a bijection, and Z is countable. ason Filippou (CMSC250 @ UMCP) Countability 06-23-2016 7 / 12
Countability of integers and rationals Z even is countable... 6 4 2 0 2 4 6... ason Filippou (CMSC250 @ UMCP) Countability 06-23-2016 8 / 12
Countability of integers and rationals Z even is countable... 6 4 2 0 2 4 6...... 3 2 1 0 1 2 3... ason Filippou (CMSC250 @ UMCP) Countability 06-23-2016 8 / 12
Countability of integers and rationals Z even is countable... 6 4 2 0 2 4 6...... 3 2 1 0 1 2 3... Call this function g. ason Filippou (CMSC250 @ UMCP) Countability 06-23-2016 8 / 12
Countability of integers and rationals Z even is countable... 6 4 2 0 2 4 6...... 3 2 1 0 1 2 3... Call this function g. Is g onto? ason Filippou (CMSC250 @ UMCP) Countability 06-23-2016 8 / 12
Countability of integers and rationals Z even is countable... 6 4 2 0 2 4 6...... 3 2 1 0 1 2 3... Call this function g. Is g onto? Is g 1-1? ason Filippou (CMSC250 @ UMCP) Countability 06-23-2016 8 / 12
Countability of integers and rationals Z even is countable... 6 4 2 0 2 4 6...... 3 2 1 0 1 2 3... Call this function g. Is g onto? Is g 1-1? Therefore, g is a bijection from Z to Z even. ason Filippou (CMSC250 @ UMCP) Countability 06-23-2016 8 / 12
Countability of integers and rationals Z even is countable... 6 4 2 0 2 4 6...... 3 2 1 0 1 2 3... Call this function g. Is g onto? Is g 1-1? Therefore, g is a bijection from Z to Z even. So gof is a bijection from N to Z even (formally prove at home)! Therefore, Z even is countable. ason Filippou (CMSC250 @ UMCP) Countability 06-23-2016 8 / 12
Countability of integers and rationals Is Q + countable? Reminder: Q + = { m n, m, N, n N } Discuss it with your neighbors for a while! Jason Filippou (CMSC250 @ UMCP) Countability 06-23-2016 9 / 12
Uncountability of R Uncountability of R Jason Filippou (CMSC250 @ UMCP) Countability 06-23-2016 10 / 12
Uncountability of R Cantor s diagonal argument Famous proof by contradiction. Method known as diagonalization, or the diagonal argument. Jason Filippou (CMSC250 @ UMCP) Countability 06-23-2016 11 / 12
Uncountability of R The proof Theorem R is uncountable. By contradiction. Suppose that R is countable. This means that we can order all the reals in a list, as follows: 0.a 11 a 12 a 13... 0.a 21 a 22 a 23... 0.a 31 a 32 a 33.... Let us now create a real number r with decimal digits r i, which will be populated as follows: { 0, aii = 9 r i = a ii + 1, 0 a ii < 9 By construction, r is different from all real numbers that we listed, since it s guaranteed to be different from the i th number at the i th decimal digit, where i = 1, 2,.... Contradiction, because we assumed that we sequentially listed all the real numbers inside this very list. Therefore, R is uncountable. Jason Filippou (CMSC250 @ UMCP) Countability 06-23-2016 12 / 12