Class 26: Wrap Up

cs2102: Discrete Mathematics
University of Virginia, Fall 2016

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

Transcript

1. Class  26:  Wrap-­‐up cs2102:  Discrete  Mathematics  |  F16 uvacs2102.github.io   David  Evans   University  of  Virginia 0
2. Plan Today: Wrapping  up  the  Course! 1 Final  Exam  is  Saturday,  9am-­‐noon  (December  10) Please  verify  your  grades   are  recorded  correctly  in   collab;  any  mistakes  need   to  be  corrected  by  Friday My  office  hours  tomorrow   (Wednesday),  4-­‐5pm  (not   1-­‐2pm) Final  PS⍵ submissions:   due  11:59pm  tonight
3. 2 Milan  Bharadwaj https://www.youtube.com/watch?v=PNpBamrpgic
5. 4 From  Course  Registration  Survey:
6. Types  of  Definitions  in  cs2102 5
7. Declarative  (Natural  Language)  Definitions 6 Definition. A  proposition  is  a  statement  that  is  either  true   or  false.  (Class  1) Definition. A  set  is  well-­‐ordered with  respect  to  an   ordering  function  (e.g.,  <),  if  any  of  its  non-­‐empty  subsets   has  a  minimum element.  (Class  3) Definition.  A  formula  is  valid if  there  is  no   way  to  make  it  false.  (Class  4)
8. Declarative  (Formal)  Definitions 7 Definition.  An  integer,  ,  is  even if  there  exists  an  integer   such  that     =  2.  (Class  2)
9. Declarative  (Formal)  Definitions 8 The  power  set  of  a  set   is  the  set  of  all  subsets  of  . ∈ ⟺ ⊆ Class  19:
10. Descriptive  Definitions 9   =  (, ⊆  ×  , 5 ∈ )
11. Descriptive  Definitions 10   =  (, ⊆  ×  , 5 ∈ ) The  execution  of  a  state  machine, = (, ⊆  ×, 5 ∈ ) is  a  (possibly  infinite)  sequence  of  states,  (5 , 8 , … , : ) that: 1.  5 = 5 (it  begins  with  the  start  state) 2.  ∀ ∈ 0, 1, … , − 1  .   B  → BD8 ∈ (if   and   are   consecutive  states  in  the  sequence,  there  is  an  edge   → ∈ .
12. Constructive Definitions 11 Definition.  A  list is  an  ordered  sequence  of  objects.     A  list  is  either  the  empty  list  (),  or  the  result  of   prepend(, ) for  some  object   and  list  .
13. Set  Cardinality  Definitions 12 If   is  a  finite  set,  the  cardinality of  ,   written  ||,  is  the  number  of  elements  in  . The  cardinality of  the  set P =       ∈ ℕ ∧ <  } is  .    If  there  is  a  bijection between  two   sets,  they  have  the  same  cardinality.
14. Desirable  Definitions Why  do  we  have  so  many  kinds  of  definitions? 13
15. 14 Baseball  Game  State  Machine By  Mason  Au,  Bobby  Stephens,  and  Kevin  Warshaw Link
16. Counterexamples 15
17. Counter  Examples 16
18. Counter  Examples 17 Diagonalization  Arguments Set  of  all  languages  of  bitstrings is  uncountable
19. Ask  Anything! 18 Helen  Simecek's Logical  Operators
24. 23 Part  of  the  big  struggle  of  mathematics  is  synthesizing  all  of  the   information  in  all  of  these  ladder  rungs  into  a  coherent  world-­‐view   that  you  can  personally  scale  up  and  down  at  will.
25. Scaling  the   Ladder  of  Abstraction 24
26. Abstractions 25 ℤ int (C,  Java),  int (Python)   Mathematical  Abstraction Concrete  Program  Representation ℝ float  (Java,  Python),  double,  etc.   set set,  frozenset (Python) function function,  procedure,  method
27. Abstracting  Programs 26 Program State  Machine
28. Abstracting  Computers 27 = (, ⊆  ×  Γ →  ×  Γ  ×  ,  5 ∈ , Z[[\]^ ⊆ ) is  a  finite  set,   Γ is  finite  set  of  symbols, = {L,  R,  Halt} The  execution  of  a  Turing  Machine, = (, ⊆  ×  Γ →  ×  Γ  ×  ,  5 ∈ , Z[[\]^ ⊆ )  is  a   (possibly  infinite)  sequence  of  configurations,   5 , 8 , …, :  where   ∈ Tsil  ×    ×  List,  such  that: 1.  5 = (5 = , 5 , 5 = ) 2.  ∀ ∈ 0, 1, …, − 1  .    (B = B , B , B → BD8 = (BD8 , BD8 , BD8 )) ∈ where  transitions  are  defined  …
29. Abstractions  Abstract 28 ℒAliG ≔  ×   =    , ∈ 9 ∗, = }}
30. 29 Physical  Computers Model  Computers Physics Transistors Circuits Machine  Code Assembly  Code High-­‐Level  Program Algorithm Compiler Low-­‐Level  Program Interpreter Assembler Loader Python C ZFC  Axioms Sets Relations State  Machines Turing  Machines Algorithm Numbers
31. 30 Physical  Computers Model  Computers Physics Transistors Circuits Machine  Code Assembly  Code High-­‐Level  Program Algorithm Compiler Low-­‐Level  Program Interpreter Assembler Loader Python C ZFC  Axioms Sets Relations State  Machines Turing  Machines Algorithm Numbers Boolean  Logic
32. Minimizing   Magic 31 Its  all  magic! Physics Four  Years  Studying   Computing  at  an  Elite   Public  University Its  all   understandable! (and  I  can  do  magical  things!) Cool  Computing  Stuff
33. Course  Goal  Reminder:  Minimizing  Magic 32 Its  all  magic! Physics Cool  Computing  Stuff cs1110 cs2110 cs2150 cs2150 cs2330 cs3330 cs3102 cs4414 cs4610 cs4414 cs4414 electives From  cs4414  (Operating  Systems  rust-­‐class.org):
34. Computer  Scientist’s    Goal:  Minimize  Magic 33 Its  all  magic! Physics Cool  Computing  Stuff cs11XX cs2330 cs3330 cs3102 cs4414 cs2102 cs4414 cs4414 Mathematics cs4102 cs3102 cs2150 cs2150 cs2110 cs2102
35. Charge 34 Thank  you!
36. Charge 35 Thank  you! Final  Exam  is  Saturday,  9am-­‐noon  (December  10) Please  verify  your  grades   are  recorded  correctly  in   collab;  any  mistakes  need   to  be  corrected  by  Friday My  office  hours  tomorrow   (Wednesday),  4-­‐5pm  (not   1-­‐2pm) Final  PS⍵ submissions:   due  11:59pm  tonight