CMSC 250 - Discrete Structures Summer 2016 Jason Filippou UMCP - - PowerPoint PPT Presentation

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CMSC 250 - Discrete Structures Summer 2016 Jason Filippou UMCP - - PowerPoint PPT Presentation

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CMSC 250 - Discrete Structures Summer 2016 Jason Filippou UMCP 05-31-2016 Jason Filippou (UMCP) Discrete Structures 05-31-2016 1 / 38 Outline 1 Overview & Logistics 2 Subject of the course Short history of Discrete Mathematics As a

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CMSC 250 - Discrete Structures Summer 2016

Jason Filippou

UMCP

05-31-2016

Jason Filippou (UMCP) Discrete Structures 05-31-2016 1 / 38

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Outline

1 Overview & Logistics 2 Subject of the course

Short history of Discrete Mathematics As a Computer Scientist... As a CS-UMD student...

3 What we’ll (tentatively) cover

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Overview & Logistics

Overview & Logistics

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Overview & Logistics

Course overview

Webpage: http://cs.umd.edu/class/summer2016/cmsc250/ May 31-July 22

Expected days “lost”: July 4, 2 Midterm days (06-17, 07-08), Final day (07-22). Look at the syllabus for policy on excused absences, academic honesty, etc

Please register on Piazza! TAs: Parsa Saadatpanah, Yancy Liao. Office hours have been posted!

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Overview & Logistics

Course overview

Textbooks (recommended):

“Discrete Mathematics and Applications”, Susanna Epp, any edition ≥ 2nd. (UMD standard, expensive to buy new). “Discrete Mathematics and Applications”, Thomas Koshy, 1st edition (cheaper, more in line with our flow). Bookstore should have a small number for rentals.

Grading (subject to minor changes):

5 homework assignments: 15% (3% each) 5 quizzes: 10% (2% each) 2 in-class midterms: 20 & 25% respectively Final (comprehensive, in-class): 30%

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Overview & Logistics

Requirements

No exceptional CS / mathematical background required for the course.

Advanced highschool math material (Calculus, Probability, Set Theory) helpful, but not required.

Charlie the Unicorn requirement: All students are required to watch this 20- minute video outlining the epic saga of “Charlie the Unicorn” and submit a half-page essay on their favorite parts of the series. We will be using elements of Charlie’s story in the early parts of the course to explain aspects of “Predicate” Logic.

Figure 1: Charlie, pictured here in between Purple and Blue Unicorns, quite distressed.

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Overview & Logistics

Your Instructor

Figure 2: If I do this trip one more time

Greek-Canadian States: 2012-today PhD, CompSci

Probabilistic Graphical Models, Action Recognition, . . . Expected graduation: ???

Likes: Coffee Dislikes: Everything else

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Overview & Logistics

My school

D.I.T, NKUA (not NTUA) Crappy buildings and infrastructure, great professors. Quite strong in:

Databases / Data Mining Theory Logic

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Overview & Logistics

My hometown

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Overview & Logistics

My hometown

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Overview & Logistics

My hometown

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Overview & Logistics

My home country

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Overview & Logistics

My home country

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Subject of the course

Subject of the course

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Subject of the course

Discrete Mathematics: The big picture

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Subject of the course

Discrete Mathematics: The big picture

MATHEMATICS

DISCRETE CONTINUOUS Logic Calculus Set theory Induction Prob-Stats Optimization Functional Analysis Number Theory

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Subject of the course Short history of Discrete Mathematics

Short history of Discrete Mathematics

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Subject of the course Short history of Discrete Mathematics

Historical Overview

The history of Discrete Mathematics largely runs parallel to that

  • f Logic and Set Theory.

Logic: Ancient Greece, Medieval Middle East, 19th century “renaissance”. Set theory: Cantor’s and Dedekind’s set theory, Russel’s and Tarski’s paradoxes, G¨

  • del’s Incompleteness Theorem

Extensions in Computer Science

Computability theory Computation theory

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Subject of the course Short history of Discrete Mathematics

Ancient Greece

Thales of Miletus: First philosopher to use deductive reasoning. Euclid: Defined axioms, propositions as well as the notion of a formal proof. Authored Elements, the first collection of axioms of geometry and number theory. Aristotle: Authored Organon, with which he tried to answer the questions: What constitutes a syllogism? Which syllogisms are valid?)

Figure 3: Euclid Figure 4: Aristotle

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Subject of the course Short history of Discrete Mathematics

Medieval Middle East

Progress made on inductive (bottom-up) reasoning. Avicennian logic was the dominant paradigm. The principles of mathematical induction were laid down at that time.

Figure 5: Ibn Sina Figure 6: Not Ibn Sina

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Subject of the course Short history of Discrete Mathematics

Modern Era

Rigorous formalization of Logic.

Leibniz Boole Russell / Whitehead Peano Hilbert

Applications to binary circuits after World War II

Figure 7: George Boole Figure 8: Bertrand Russel

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Subject of the course Short history of Discrete Mathematics

Set Theory

Axiomatization of set theory in the late 19th century.

Cantor & Dedekind (Cantorian set theory). Russel’s Paradox.1 Hilbert’s Hotel.

Limitations of the algorithmic procedure.

G¨

  • del’s Incompleteness TheoremTM

Tarski’s Undefinability TheoremTM The halting problem. Figure 9: Georg Cantor Figure 10: Alfred Tarski

1Independently and simultaneously discovered by Ernst Zermelo. Jason Filippou (UMCP) Discrete Structures 05-31-2016 22 / 38

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Subject of the course As a Computer Scientist...

As a Computer Scientist...

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Subject of the course As a Computer Scientist...

Where Discrete Math fits (in CS)

Mathematical backbone for Theory!

Counting and probability paramount! Inductive proofs of correctness everywhere.

Applications of logic

“Vanilla” logic, DataLog and deductive databases. Probabilistic logics (e.g MLNs) and graph databases. Automated theorem provers (commercial / academic prototypes)

Set Theoretical elements paramount for:

Computability theory. Study of compilers.

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Subject of the course As a Computer Scientist...

But I just want to code!

So you want to be hired by a software company. 4 to 5 interviews. First 2 questions in 1st - 2nd interviews are usually low-level theoretical and may contain examples such as:

Among the residents of [insert name of city that you’re interviewing for a position at], is it possible that you can find two people with the exact same number of hairs on their head? If I have a full binary tree of height 10 and I add another level of leaves to it, how many nodes will I have total?

First question is an application of the Pigeonhole Principle. Second question requires an inductive proof as a step in the answer (some might argue 2 inductive proofs, one classic and one structural).

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Subject of the course As a CS-UMD student...

As a CS-UMD student...

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Subject of the course As a CS-UMD student...

Where Discrete Structure fits (in the curriculum)

216 250

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Subject of the course As a CS-UMD student...

Where Discrete Structure fits (in the curriculum)

216 250 351

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Subject of the course As a CS-UMD student...

Where Discrete Structure fits (in the curriculum)

216 250 351 (Also: 330, 320,...)

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Subject of the course As a CS-UMD student...

Where Discrete Structure fits (in the curriculum)

216 250 351 (Also: 330, 320,...) 421 430

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What we’ll (tentatively) cover

What we’ll (tentatively) cover

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What we’ll (tentatively) cover

Part 1

Logic (Weeks 1 & 2).

Propositional logic. Applications on Boolean Circuits. “Predicate” logic.

Formal proof methodology (Weeks 2 & 3).

Existential proofs. Constructive proofs. Proofs by contradiction.

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What we’ll (tentatively) cover

1st Midterm!

Friday, 06-17. In-class, 85 minutes.

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What we’ll (tentatively) cover

Part 2

Number theory (Week 3, intertwined with proofs).

Prime and Composite numbers, relevant theorems. Divisibility. Modular Arithmetic. Fundamental Theorem of ArithmeticTM

Set and Function theory (Week 4).

Basic axioms and properties. Proofs on sets. Function definitions (injective, reflexive, bijective, onto...). Countable and uncountable sets.

Induction (Weeks 5-6)

Weak induction. Strong induction. Constructive induction. Strings, Trees, Graphs and Structural induction.

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What we’ll (tentatively) cover

2nd Midterm!

Friday, 07-08. In-class, 85 minutes.

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What we’ll (tentatively) cover

Part 3

Counting and Probability (Weeks 7-8)

Basic series and sums. Permutations. Combinations, r-combinations. Events, Venn Diagrams and Probability. Sum and product rules.

Open lectures. Possibilities:

Counting beyond infinity (Hilbert’s Hotel, Ordinals, Cardinals, Aleph-n sets, Continuum Hypothesis...) Relations. Recursion and classic recursive algorithms. Guest speaker.

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What we’ll (tentatively) cover

Final!

Friday, 07-22. In-class, comprehensive, 110 minutes.

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What we’ll (tentatively) cover

Questions?

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