Countability. Jason Filippou UMCP. Jason Filippou UMCP) Countability / 12

Transcription

1 Countability Jason Filippou UMCP Jason Filippou UMCP) Countability / 12

2 Outline 1 Infinity 2 Countability of integers and rationals 3 Uncountability of R Jason Filippou UMCP) Countability / 12

3 Infinity Infinity Jason Filippou UMCP) Countability / 12

4 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 UMCP) Countability / 12

5 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 UMCP) Countability / 12

6 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 UMCP) Countability / 12

7 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 UMCP) Countability / 12

8 Countability of integers and rationals Countability of integers and rationals Jason Filippou UMCP) Countability / 12

9 Z is countable Countability of integers and rationals Jason Filippou UMCP) Countability / 12

10 Z is countable Countability of integers and rationals Jason Filippou UMCP) Countability / 12

11 Z is countable Countability of integers and rationals Let s call this function f. We can make the following observations about f: Jason Filippou UMCP) Countability / 12

12 Z is countable Countability of integers and rationals 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 UMCP) Countability / 12

13 Z is countable Countability of integers and rationals Let s call this function f. We can make the following observations about f: 1 No integer is counted twice! So, f is...? All integers are (eventually) accounted for! So, f is...? ason Filippou UMCP) Countability / 12

14 Z is countable Countability of integers and rationals Let s call this function f. We can make the following observations about f: 1 No integer is counted twice! So, f is...? All integers are (eventually) accounted for! So, f is...? onto. ason Filippou UMCP) Countability / 12

15 Z is countable Countability of integers and rationals Let s call this function f. We can make the following observations about f: 1 No integer is counted twice! So, f is...? 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 UMCP) Countability / 12

16 Countability of integers and rationals Z even is countable ason Filippou UMCP) Countability / 12

17 Countability of integers and rationals Z even is countable ason Filippou UMCP) Countability / 12

18 Countability of integers and rationals Z even is countable Call this function g. ason Filippou UMCP) Countability / 12

19 Countability of integers and rationals Z even is countable Call this function g. Is g onto? ason Filippou UMCP) Countability / 12

20 Countability of integers and rationals Z even is countable Call this function g. Is g onto? Is g 1-1? ason Filippou UMCP) Countability / 12

21 Countability of integers and rationals Z even is countable Call this function g. Is g onto? Is g 1-1? Therefore, g is a bijection from Z to Z even. ason Filippou UMCP) Countability / 12

22 Countability of integers and rationals Z even is countable 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 UMCP) Countability / 12

23 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 UMCP) Countability / 12

24 Uncountability of R Uncountability of R Jason Filippou UMCP) Countability / 12

25 Uncountability of R Cantor s diagonal argument Famous proof by contradiction. Method known as diagonalization, or the diagonal argument. Jason Filippou UMCP) Countability / 12

26 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 a 21 a 22 a a 31 a 32 a 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 UMCP) Countability / 12