CMSC 250 - Discrete Structures Summer 2016 250 - Discrete Structures Summer 2016 ... Ancient Greece, Medieval Middle East, ... Subject of the course Short history of Discrete Mathematics

CMSC 250 - Discrete StructuresSummer 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 courseShort history of Discrete MathematicsAs a Computer Scientist...As a CS-UMD student...

3 What we’ll (tentatively) cover


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), Finalday (07-22).Look at the syllabus for policy on excused absences, academichonesty, etc

Please register on Piazza!

TAs: Parsa Saadatpanah, Yancy Liao.

Office hours have been posted!


http://cs.umd.edu/class/summer2016/cmsc250/

http://cs.umd.edu/class/summer2016/cmsc250/

http://piazza.com/umd/summer2016/cmsc250


Course overview

Textbooks (recommended):

“Discrete Mathematics and Applications”, Susanna Epp, anyedition ≥ 2nd. (UMD standard, expensive to buy new).“Discrete Mathematics and Applications”, Thomas Koshy, 1stedition (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% respectivelyFinal (comprehensive, in-class): 30%



Requirements

No exceptional CS / mathematical background required for thecourse.

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

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

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


https://www.youtube.com/watch?v=fZu3P0OsQPk


Your Instructor

Figure 2: If I do this trip one more time

Greek-Canadian

States: 2012-today

PhD, CompSciProbabilistic Graphical Models, Action Recognition, . . .Expected graduation: ???

Likes: Coffee

Dislikes: Everything else



My school

D.I.T, NKUA (not NTUA)

Crappy buildings and infrastructure, great professors.Quite strong in:

Databases / Data MiningTheoryLogic



My hometown



My hometown



My hometown



My home country



My home country


Subject of the course




Discrete Mathematics: The big picture



Discrete Mathematics: The big picture

MATHEMATICS

DISCRETE CONTINUOUS

Logic

Calculus

Set theory

Induction

Prob-Stats

Optimization

FunctionalAnalysis

Number Theory


Subject of the course Short history of Discrete Mathematics

Short history of Discrete Mathematics



Historical Overview

The history of Discrete Mathematics largely runs parallel to thatof Logic and Set Theory.

Logic: Ancient Greece, Medieval Middle East, 19th century“renaissance”.

Set theory: Cantor’s and Dedekind’s set theory, Russel’s andTarski’s paradoxes, Godel’s Incompleteness Theorem

Extensions in Computer Science

Computability theoryComputation theory



Ancient Greece

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

Figure 3: Euclid Figure 4: AristotleJason Filippou (UMCP) Discrete Structures 05-31-2016 19 / 38


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 thattime.

Figure 5: Ibn Sina Figure 6: Not Ibn Sina



Modern Era

Rigorous formalization of Logic.LeibnizBooleRussell / WhiteheadPeanoHilbert

Applications to binary circuits after World War II

Figure 7: George BooleFigure 8: Bertrand Russel



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.Godel’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

Subject of the course As a Computer Scientist...

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.



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-leveltheoretical and may contain examples such as:

Among the residents of [insert name of city that you’re interviewingfor a position at], is it possible that you can find two people withthe exact same number of hairs on their head?If I have a full binary tree of height 10 and I add another level ofleaves 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 theanswer (some might argue 2 inductive proofs, one classic and onestructural).


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

As a CS-UMD student...



Where Discrete Structure fits (in the curriculum)

216250




216250

351




216250

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




216250

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

421 430


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.



1st Midterm!

Friday, 06-17.

In-class, 85 minutes.



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.



2nd Midterm!

Friday, 07-08.

In-class, 85 minutes.



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.



Final!

Friday, 07-22.

In-class, comprehensive, 110 minutes.



Questions?


Documents

CMSC 250 - Discrete Structures Summer 2016 250 - Discrete Structures Summer 2016 ... Ancient Greece, Medieval Middle East, ... Subject of the course Short history of Discrete Mathematics