Class 21: Infinite Infinities, Exam 2

cs2102: Discrete Mathematics
University of Virginia, Fall 2016

See course site for notes:
https://uvacs2102.github.io

Transcript

1. Class  21:   Seeing  is   Believing! cs2102:  Discrete  Mathematics  |  F16 uvacs2102.github.io   0 David  Evans   University  of  Virginia
2. Plan Today:  Infinite  Cardinalities – Uncountable  Sets,  Cantor’s  Proof;  Exam  2 Thursday:  Uncomputable Numbers (Turing) Rest  of  semester: – Slack  poll  on  potential  topics – Other  suggestions  welcome 1 PS9  due  next  Wednesday
3. Recap:  Definitions 2 A  set   is  countable  if  and  only  if   there  exists  a  surjective  function from  ℕ to  . A  set   is  infinite,  if  there  is  no   bijection  between   and   any  ℕ% . A  set   is  countably  infinite  if  it  is  countable and  it  is  infinite. A  set    is  countably  infinite  if  there  exists  a  bijection   between   and  ℕ.
4. Countable  Cists? 3
5. Cardinality  of  cists  =  |ℕ| 4
6. Cantor’s  Theorem 5 For  all sets  ,   > |    |.
7. Cardinality  of  Reals  in  [0, 1) 6
8. Cantor’s  Diagonalization  Argument 7
9. What’s  bigger?   ℕ    ?  |         ∈  ℝ, 0 ≤ < 1}   8
10. Is  there  anything  bigger? 9
11. Exam  2 10
12. Problem  2c 11 ≤ 1   ≤ 1  
13. Preserved  Invariant 12
14. Preserved  Invariant? 13
17. 16 Why  prove  about  state  machine  that  models  code  and  not  about  code?
18. 17 Termination
19. 18 Correctness
21. Question  Breakdown 20 1 2 3 4 5 6 7 8 Num great   (>=10;  #2 >=  9;  #3  >=  12) 39% 9% 17% 28% 19% 35% 26% 25% Num passed   (>=8; #2  >=  7;  #3  >=  10) 46% 23% 31% 28% 28% 42% 43% 38% Num okay (>=6) 52% 45% 44% 48% 41% 53% 53% 50%
22. Returning  Exam  1 Group  1 Group  2 Group 3 Group  4 aa6..  (Aaron) dav..  (Daniel) kaw..  (Kevin)   rcb..  (Rachel) ac5..  (Allison) dc7..  (David) khl..  (Kenny) rel..  (Rashid) … … … … cwf… (Chris) jxp..  (John) pv..  (Phoenix) zw4..  (Mike)
23. Charge Infinities  are  Spooky,  Strange,  and  Surprising PS9  due  next  Wednesday   (will  be  posted  tomorrow) 22