Countability Jason Filippou CMSC250 @ UMCP 06-23-2016 Jason - PowerPoint PPT Presentation

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

Infinity Infinite sets 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

Infinity Countable sets 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

Infinity Countable sets 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

Infinity Countable sets 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

Countability of integers and rationals Z is countable . . . − 3 − 2 − 1 0 1 2 3 . . . Jason Filippou (CMSC250 @ UMCP) Countability 06-23-2016 7 / 12

Countability of integers and rationals Z is countable . . . − 3 − 2 − 1 0 1 2 3 . . . . . . 7 5 3 1 2 4 6 . . . Jason Filippou (CMSC250 @ UMCP) Countability 06-23-2016 7 / 12

Countability of integers and rationals Z is countable . . . − 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

Countability of integers and rationals Z is countable . . . − 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

Countability of integers and rationals Z is countable . . . − 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... ? Jason Filippou (CMSC250 @ UMCP) Countability 06-23-2016 7 / 12

Countability of integers and rationals Z is countable . . . − 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 . Jason Filippou (CMSC250 @ UMCP) Countability 06-23-2016 7 / 12

Countability of integers and rationals Z is countable . . . − 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. Jason 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 . . . . . . Jason 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 . . . Jason 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 . Jason 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? Jason 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? Jason 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 . Jason 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. Jason 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 , a ii = 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