Class 26: Wrap Up - Speaker Deck

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Class 26: Wrap-‐up cs2102: Discrete Mathematics | F16 uvacs2102.github.io David Evans University of Virginia 0

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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

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2 Milan Bharadwaj https://www.youtube.com/watch?v=PNpBamrpgic

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4 From Course Registration Survey:

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Types of Definitions in cs2102 5

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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)

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Declarative (Formal) Definitions 7 Definition. An integer, , is even if there exists an integer such that = 2. (Class 2)

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Declarative (Formal) Definitions 8 The power set of a set is the set of all subsets of . ∈ ⟺ ⊆ Class 19:

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Descriptive Definitions 9 = (, ⊆ × , 5 ∈ )

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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 → ∈ .

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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 .

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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.

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Desirable Definitions Why do we have so many kinds of definitions? 13

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14 Baseball Game State Machine By Mason Au, Bobby Stephens, and Kevin Warshaw Link

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Counterexamples 15

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Counter Examples 16

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Counter Examples 17 Diagonalization Arguments Set of all languages of bitstrings is uncountable

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Ask Anything! 18 Helen Simecek's Logical Operators

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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.

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Scaling the Ladder of Abstraction 24

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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

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Abstracting Programs 26 Program State Machine

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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 …

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Abstractions Abstract 28 ℒAliG ≔ × = , ∈ 9 ∗, = }}

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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

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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

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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

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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):

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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

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Charge 34 Thank you!

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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