discrete mathematics introductionansaf/courses/3203/3203_lecture1_intro.pdf · this class is an...
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Discrete Mathematics
Introduction
Ansaf Salleb-AouissiColumbia University
COMS W3203 Discrete Mathematics
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Outline
1. Instruction team
2. Course logistics
(a) Syllabus
(b) Textbook
(c) Grading criteria
3. What is Discrete Math?
4. Course content
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Canvas and Piazza
We will use:
• Canvas (courseworks) for the official announcements.
• Piazza as a discussion forum. Don’t be shy for asking! You
can also post privately.
• Gradescope and canvas for HW submission.
• No website.
• Syllabus. Please read.
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Homework
1. Explain, Proof-read your work.
2. LATEX it: We want to see your HW returned in LATEX.
3. Homework 1 on LaTeX: required, and counts as one homework.
4. LaTeX resources:
(a) Using Overleaf/ShareLaTeX
(b) Installing a TEX distribution in your local machine
(c) LaTeXiT
(d) Science & Engineering Library
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Academic honestyAcademic integrity will be strictly enforced.
• Columbia University Guide to Academic Integrity:
https://www.college.columbia.edu/academics/
academicintegrity
• Department of Computer Science Academic Honesty Pol-
icy:
http://www.cs.columbia.edu/education/honesty
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Textbook
Textbook
1. Mathematics: a discrete introduction. Scheinerman, Edward
R. BrooksCole, 2013. Third Edition.
2. Full of practice questions with solutions or hints of solutions
at the end.
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Introduce yourself!
• Please introduce yourself to your neighbors.
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What is this class about?
• This class is an introductory class in Discrete Mathematics
with two primary goals:
1. Teach fundamental discrete math concepts.
2. Teach how to write proofs – How to think and write clearly.
• You will see most of the topics covered again/used in later CS
courses. This course introduces them.
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Testing out
• For some of you, the topics will be completely new, others not.
• If for you, it is a lot of deja-vu, like 80% of the course content,
consider testing out.
• See my announcement for more information.
• This class is meant to start from scratch.
• Statistics from previous semesters.
• “I am good in math already, so I should do well!” Let’s talk
about this...
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What is Discrete Math?
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What is Discrete Math?
• Mathematics can be roughly divided into discrete math (DM)
and continuous math (CM).
• Analogy: DM is similar to a digital watch, only discrete time
is displayed (where there is no split second).
• CM is similar to an analog watch displaying a continuous time.
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What is Discrete Math?
• DM deals with integers, puzzles, proof writing and induction.
• CM deals with real numbers to model real world phenomenon
along with notions like continuity, derivatives, limits, differen-
tial equations, etc.
• CM is older than DM
• DM flourished in the era of computers and has been very use-
ful in applications such as scheduling airlines, communication,
crypto systems, encoding movies and songs, databases, secu-
rity, computer networks, etc.
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Course content• Logic: Propositional logic, truth tables, Boolean algebra, inference.
• Proofs: if-then proof, contradiction, by cases, counter example
• Collections (set theory) Lists, sets, operations, factorial, cardinality,quantifiers
• Relations Relations, equivalence, partitions
• More proofs Smallest counter example, proof by induction
• Counting Multiplication theorem, Anagrams, permutations, binomial co-efficients, Pascal triangle, Inclusion-Exclusion
• Functions Functions, properties (injection, surjection, bijection), compo-sition, inverse
• Graph Theory graphs, degrees, handshaking theorem, trees, planar graphs.
• Number Theory Dividing, Greatest common divisor, modular arithmetic,prime numbers, RSA public key encryption.
• Probability Sample space, events, random variables, independence, Bayesrule, conditional probability, expectation, variance.
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Course content
Most of the topics are essential for applications
in computer science and engineering
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Logic
• Emphasis: Logical thinking and mathematical notation. You
will learn:
– Use formal symbols in propositional logic.
– Find the truth value of an expression/statement.
– Make inference.
• Keywords: Propositional logic, truth tables, Boolean algebra,
theorems, truth, circuits, proofs, inference.
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Logic• Teasers:
– It is sunny and 91F. How to interpret this statement?
– It is not cold. How to interpret this statement?
– Salad or fries come with the chicken. How to interpret this
statement?
– If you know how to swim, then you will be hired as a life-
guard.
– If you know how to swim, then you will be hired as a life-
guard. You know how to swim, therefore you will be hired
as a life-guard.
– Let’s play Wumpus!
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Proofs
• So you learned how to write mathematical statement in logic.
• Then, you will learn how to:
– think and write clearly, that is write a mathematical essay
that shows without doubt that a statement is True.
– use different proof techniques.
• keywords: if-then proof, proof by contradiction, by cases,
counter example.
• Teaser: For every integer n, n2 + n + 41 is a prime number.
Is this true?
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Collections
• You will learn how to deal with lists or collections of objects
and how to count them.
• keywords: Lists, sets, operations, factorial, cardinality, quanti-
fiers.
• Teaser:
- What is in your backpack?
- How many license plates are possible if we use six characters,
the first 3 are upper case and the last 3 are digits 0-9.
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Relations
• You will learn how to connect things with relationships, their
properties, equivalence relations and equivalence classes.
• keywords: Relations, equivalence, partitions, equivalence
classes.
• Teaser: The ’=’ on integers is a relation of equality:
- Reflexive: any integer is equal to itself
- Symmetric
- Transitive
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More proofs
• You will learn how to do more proofs!
• keywords: Smallest counter examples and proof by induction.
• Teaser:
- Chain reaction of dominos.
- Induction machine
Induc&on(Machine(
1+3+5=9'
1+3+5+7=9+7=16'
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Counting
• Whenever we have a question: How many ...? we are dealing
with a counting problem.
• You will learn how to count, e.g., the number of k-element
subsets if an n-element set (choose). Crucial in Probability!
• keywords: binomial coefficients, pascal triangle, inclusion-
exclusion, counting multisets
• Teaser:
How many ways there are to choose 4 courses out of 10 pos-
sibilities?(104
)= 210
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Functions
• You will learn how functions also play an important role in
discrete math.
• keywords: Discrete functions, properties, composition
• Teaser: Pigeonhole principle
In a class of 163 students, must at least two or more have the
same birthday?
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Functions
• You will learn how functions also play an important role in
discrete math.
• keywords: Discrete functions, properties, composition
• Teaser: Pigeonhole principle
In a class of 163 students, must at least two or more have the
same birthday?
No. There are 366 possible birthday. Each student could have
a distinct birthday. (there are more holes than pigeons)
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Number Theory
• You will learn how the oldest branches of mathematics is cen-
tral in the world of cryptography and todays computer security.
• Keywords: Dividing, Greatest common divisor, modular arith-
metic, prime numbers, RSA public key encryption.
• Teaser:
Every even integer greater than 2 can be expressed as the sum
of 2 primes.
Example: 24 = 11 + 13
Is this true?
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Number Theory
• You will learn how the oldest branches of mathematics is cen-
tral in the world of cryptography and todays computer security.
• Keywords: Dividing, Greatest common divisor, modular arith-
metic, prime numbers, RSA public key encryption.
• Teaser:
Every even integer greater than 2 can be expressed as the sum
of 2 primes.
Example: 24 = 11 + 13
Is this true?
An unsolved mystery: Golbach conjecture 1742!
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Graph Theory
• You will learn how to represent relationships with graphs of
vertices and edges. Very useful to model many problems in CS
and engineering.
• Keywords: graphs, vertex, edge, degree, tree, planar graph,
connectivity.
• Teaser: The problem of seven bridges: (textbook page 333)
Is there a tour we can take over the city so as we traverse each
bridge only once?
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Graph Theory
• Teaser:
Source: Image from Wikipedia
The four color problem.
The regions in any map can be colored in 4 colors so as adja-
cent regions have different colors.
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Probability
• You will learn how to cast counting problem in the language of
probability. In life, we like to analyze how likely things happen.
• keywords: Sample space, events, random variables, distribu-
tion, independence, Bayes, conditional probability, expectation,
variance.
• Teaser: PowerBall Lottery: Five white balls 1 to 69 red power-
ball 1 to 26. The jackpot - won by matching all five white balls
in any order and the red Powerball worth $1.6B.
1. How many different outcomes are there for the five white
balls (if order doesn’t matter)?
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By next lecture...
Please read Appendix D in the textbook for a list
of what we will assume in this class.
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Credit• Mathematics: a discrete introduction. Scheinerman, Edward
R. 2013
• Wikipedia for the 4 color problem picture.
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Thank you
Questions?
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