Class 27: Conclusion cs2102: Discrete Mathematics | F17 uvacs2102.github.io David Evans University of Virginia Lawn Lighting 2001

Goldilocks and the Three Boolean Bears Arjun Iyer, Abhishek Shinde, Chiraag Umesh, and Vineeth Gaddam https://www.youtube.com/watch?v=d4r_oX1AV_c

Plan Today: Wrapping up the Course! 3 Final Exam is Thursday, 9am-noon (December 7)

4 Registration survey:

What’s on the Final Exam? Devin Kim https://www.youtube.com/watch?v=6Js87ft5GKk

Well-Ordering Mystery Eimara Mirza https://www.youtube.com/watch?v=1JMJ71MbHsQ

Well-Ordering Principle Mia Shaker https://www.youtube.com/watch?v=ltj7PDWYnMM

8 Registration survey:

9 Registration survey:

I'm good at this already 4% I can do this already, but want to get better 64% I do not have this habit now 31% 10 From Course Registration Survey: Discussing Definitions

Types of Definitions in cs2102 11

Galois Rap Lindsey Shavers, Danielle Newman, Esther Boachie https://www.youtube.com/watch?v=VT0IdZrRHd0

Declarative (Natural Language) Definitions 13 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)

Declarative (Formal) Definitions 14 Definition. An integer, , is even if and only if there exists an integer such that = 2. (Class 2)

Descriptive Definitions 15 = (, ⊆ × , . ∈ )

State Machines Surbhi Singh, Raghava Pamula, Anoop Sana https://www.youtube.com/watch?v=kbGnVl53Rm0

Descriptive Definitions 17 = (, ⊆ × , . ∈ ) The execution of a state machine, = (, ⊆ ×, . ∈ ) is a (possibly infinite) sequence of states, (. , 2 , … , 4 ) that: 1. . = . (it begins with the start state) 2. ∀ ∈ 0, 1, … , − 1 . < → <>2 ∈ (if and are consecutive states in the sequence, there is an edge → ∈ .

Using the Definition = ( = ℕ, = → A > 0 ∧ (A = − 1 ∨ A = }, . = 2102) Does terminate?

Using the Definition = ( = ℕ, = → A > 0 ∧ (A = − 1 ∨ A = }, . = 2102) Does terminate? A state machine terminates if all executions of that machine eventually reach a state with no transitions.

= ( = ℕ, = → A > 0 ∧ (A = − 1 ∨ A = }, . = 2102) Does terminate?

Constructive Definitions 21 UnaryNumber =

Recursive Definitions 22 Value(UnaryNumber) :=

Galois Rap Megha Batra, Cathy Chang https://youtu.be/WBKhCF1Qo8A

24 Jeremy Kun’s essay from Course Registration survey:

25 I'm good at this already, 42 I can do this already, but want to get better, 213 I do not have this habit now, 59 0 From Course Registration Survey: Coming up with Counterexamples

Counter Examples 26

Counter Examples 27

Counter Examples 28 Diagonalization Arguments Set of all languages of bitstrings is uncountable

29 Jeremy Kun’s essay from Course Registration survey:

30 The mathematical habit is putting your personal pride or embarrassment aside for the sake of insight. I'm good at this already, 85 I can do this already, but want to get better, 169 I do not have this habit now, 59 Being Wrong

Mathematical Music Theory Makonnen Makonnen https://www.youtube.com/watch?v=lpS6KJ_-M5k

32 Can a field have more than one additive identity?

33

34 Jeremy Kun’s essay from Course Registration survey:

35 I'm good at this already, 9 I can do this already, but want to get better, 120 I do not have this habit now, 153 I didn't understand what this means, 31 From Course Registration Survey: Scaling the ladder of abstraction 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.

Scaling the Ladder of Abstraction 36

Abstractions 37 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

Abstracting Programs 38 Program State Machine

39 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

Minimizing Magic 40 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

Computer Scientist’s Goal: Minimize Magic 41 Its all magic! Physics Cool Computing Stuff cs11XX cs2330 cs3330 cs3102 cs4414 cs2102 cs4414 cs4414 Mathematics cs4102 cs3102 cs2150 cs2150 cs2110 cs2102

No content

Discrete Song Max Rifkin https://soundcloud.com/max-rifkin/discrete-song-copy

Start with a proposition that maybe true or false and then assume your axioms to base other things off. Make sure your rules are labeled sound if they only return true. If you're not sure make a table, they always show the truth. We learned of operators ANDs and ORs and P’s and Q’s, we talked of implications and when they will be true. Dave taught us contrapositives and how to write good proofs. We then tackled well ordering and we were all confused. With respect to some relation in non-empty subsets, there's got to be a minimum or order is upset. These proofs use contradiction and counter example sets, assuming it's well ordered too is probably your best bet. There's CNF and DNF and then the DeMorgan's law NANDs and NORs exclusive ORs we all learned in the fall. Binary relations, graphs of edges out and in Got to use all your resources cause your head might start to spin. Go to office hours the diligent and kind the TAs they will guide you help you make up your mind. If you've got no idea and you're ready to give in the lectures that are on collab are probably your best friend To use proof by induction first think of your base case , and if your proposition holds well boy you're in the race the easy part is over now prove p(n+1) by assuming you've got p(n) if you can then you’re done Discrete Song by Max Rifkin

Charge 45 Thank you! Final Exam is Thursday, 9am-noon Please verify your grades are recorded correctly in collab; any mistakes need to be corrected by Friday I will have my normal office hours tomorrow (2:30-3:30pm), and normal TA office hours tomorrow. Last scheduled office hours.