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CSci 2011 Discrete Mathematics Lecture 1 IntroductionPowerPoint Presentation

CSci 2011 Discrete Mathematics Lecture 1 Introduction

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Instructor, TA, Office Hours

- Instructor
- Yongdae Kim (Fourth time teaching 2011)
- Email: kyd(at)cs. umn. edu
- Please include 2011 in the subject of your mail

- Office: 200 Union St. SE, EECS Building (Keller Hall), room 4-225E
- Office Hours: T 11:00 ~ 12:00, Th 10:00 ~ 11:00 (Also by appointment)

- Teaching Assistants
- Ben Dischinger, disch029(at)umn.edu, MW 9:45 AM - 10:45 AM
- Abedelaziz Mohaisen (Aziz), mohaisen(at)cs.umn.edu, Th 11:30 AM - 1:30 PM
- Jeremy Iverson, jiverson(at)cs.umn.edu, Tu 2:30 PM - 3:30 PM
- Shaun Goss, goss0063(at)umn.edu, Th 2:30 PM - 3:30 PM
- Nathan Fox, foxxx340(at)umn.edu, Th 1:30 PM - 2:30 PM
- Katie Wolf, wolfx265(at)umn.edu, W 1:30 PM - 2:30 PM

- Recitation
- Section 001: Ben (Lead), Katie, Shaun
- Section 002: Ben (Lead), Aziz, Katie
- Section 003: Aziz (Lead), Ben, Nathan
- Section 004: Aziz (Lead), Jeremy, Shaun
- Section 005: Jeremy (Lead), Shaun, Nathan

CSci 2011

Class web page, e-mail

- http://www-users.itlabs.umn.edu/classes/Fall-2010/csci2011/
- Reading the page carefully and regularly!
- Read the Syllabus carefully.
- Check calendar.

- E-mail policy
- Include [2011] in the subject of your e-mail
- Use TA as much as possible :-)

CSci 2011

Overview

- Much of the basic mathematical machinery useful in computer science will be presented, with applications.
- Students will learn actively the art of creating real-world proofs in these areas,
- preparing them for diverse regions of computer science such as architecture, algorithms, automata, programming languages, cryptography, and
- increasing their general problem-solving abilities in all areas.

CSci 2011

Problems solved using Discrete Math

- How many secure passwords?
- Probability of winning Texas Hold’em?
- How can I encrypt a message?
- Shortest paths between two cities using public transportation?
- How many steps required to sort 10,000 numbers? Is this algorithm correct?
- How to design a circuit that multiply two integers?

CSci 2011

Why study Discrete Maths?

- Proof
- Ability to understand and create mathematical argument

- Gateway to more advanced CS courses
- Data structures, algorithms, automata theory, formal languages
- Database, networks, operating system, security

CSci 2011

Guide for Successful Study

- No minimalist approach
- Homework would be sufficient! NOPE!!!
- Read relevant sections before coming to class
- Do the homework (of course!!!)
- Solve much more problems (odd numbered)

- Work regularly
- Most chapters are building blocks for other chapters
- So you cannot catch up 2 week lectures in 2 days

- On average 10 hours EVERY week!

- Most chapters are building blocks for other chapters
- Creativity
- No questions will require you to put just numbers to formula.
- Need to know how to apply! This can be improved by practice!

- Learning Book, class, note, homework
- It is combination of everything!

- Think yourself, discuss with your friends, write your own answer!

CSci 2011

Course content

very approximately in temporal order

- Ch. 1: Logic and Proofs
- Ch. 2: Sets, Functions, Sequences and Sums
- Ch. 3: Algorithms, the Integers, and Matrices
- Ch. 4: Induction and Recursion
- Ch. 5: Counting
- Ch. 6: Discrete Probability
- Ch. 8: Relations
- Ch. 12: Modeling Computation

CSci 2011

Typical Schedule

- Tuesday
- Lecture: 75 minutes
- Group Work Due (Given in recitation section, every week)

- Thursday
- Lecture: 75 minutes
- Assignment due (Every other week)
- Posted on Sunday (1.5 week is given)
- Topics covered until the Tuesday in the same week

- Quiz: Every other Wednesday (50 min)

- Wednesday
- Recitation: 50 minutes
- Group assignment. (Formed by instructor)
- Due: next Tuesday

- Recitation: 50 minutes

CSci 2011

Evaluation (IMPORTANT!)

- The following rules will be strictly enforced.
- Evaluation:
- Assignments (6), group assignments (12), quizzes (6), and a Final exam.
- You must pass every quiz individually by attaining at least 50% of the available points on each
- Students who fail more than once will receive an F for the course.
- All quizzes and examinations are closed book and closed notes. (One page cheat sheet is OK.)
- Do not schedule any absence (especially on Thursday) during the semester - there are no make-up quizzes.

CSci 2011

Due dates and Submission

- Due dates for all assignments are strict
- All assignments must be received at the very start of the class to receive credit.
- No late assignment will be graded.

- Keep a copy of each of your submissions as evidence that you have indeed submitted each assignment.
- Do not ever put your assignment under the instructor’s office door.

CSci 2011

Grading

- Absolute (i.e. not on a curve).
- The overall grade will be based upon
- 3% for each homework, 1% for each group assignment, 7% for each quiz, and 28% for the final.
- A minimum of 60% is necessary for an S or C- grade.

- Grading will be as follows
- 95.0% or above yields an A, 90.0% an A-
- 85% = B+, 80% = B, 75% = B-
- 70% = C+, 65% = C, 60% = C-
- 55% = D+, 50% = D, and less than 50% yields an F.

- Percentages are not rounded when using this scheme.
- Extra credit questions will be always available.

CSci 2011

Grading questions and Complaints

- Grading is performed by the TAs.
- If you have a question about grading, talk to TAs.
- Only if something unreasonable has occurred will the instructor intervene.

- Furthermore, there is a limit of ten days from when an assignment or quiz is returned in recitation (whether you are there to receive it or not) for grading problems to be dealt with.
- After that period, such will not be considered.
- The sole exception to this rule is the final examination.

CSci 2011

And…

- Incompletes (or make up exams) will in general not be given.
- Exception: a provably serious family or personal emergency arises with proof and the student has already completed all but a small portion of the work.

- Scholastic conduct must be acceptable. Specifically, you must do your assignments, quizzes and examinations yourself, on your own.

CSci 2011

Propositions

- A proposition is a statement that can be either true or false
- “Yongdae has an Apple laptop.”
- “Yongdae is a professor.”
- “3 = 2 + 1”
- “3 = 2 + 2”

- Not propositions:
- “Are you Bob?”
- “x = 7”
- “I am heavy.”

CSci 2011

Propositional variables

- We use propositional variables to refer to propositions
- Usually are lower case letters starting with p (i.e. p, q, r, s, etc.)
- A propositional variable can have one of two values: true (T) or false (F)

- A proposition can be…
- A single variable: p
- An operation of multiple variables: p(qr)

CSci 2011

Introduction to Logical Operators

- About a dozen logical operators
- Similar to algebraic operators + * - /

- In the following examples,
- p = “Today is Friday”
- q = “Today is my birthday”

CSci 2011

Logical operators: Not

- A “not” operation switches (negates) the truth value
- Symbol: or ~
- p = “Today is not Friday”

CSci 2011

Logical operators: And

- An “and” operation is true if both operands are true
- Symbol:
- It’s like the ‘A’ in And

- pq = “Today is Friday and today is my birthday”

CSci 2011

Logical operators: Or

- An “or” operation is true if either operands are true
- Symbol:
- pq = “Today is Friday or today is my birthday (or possibly both)”

CSci 2011

Logical operators: Conditional 1

- A conditional means “if p then q”
- Symbol:
- pq = “If today is Friday, then today is my birthday”
- p→q=¬pq

the

antecedent

the

consequence

CSci 2011

Logical operators: Conditional 2

- Let p = “I am elected” and q = “I will lower taxes”
- I state: p q = “If I am elected, then I will lower taxes”
- Consider all possibilities
- Note that if p is false, then the conditional is true regardless of whether q is true or false

CSci 2011

Logical operators: Conditional 3

- Alternate ways of stating a conditional:
- p implies q
- If p, q
- p only if q
- p is sufficient for q
- q if p
- q whenever p
- q is necessary for p

CSci 2011

Logical operators: Conditional 4

CSci 2011

Logical operators: Bi-conditional 1

- A bi-conditional means “p if and only if q”
- Symbol:
- Alternatively, it means “(if p then q) and (if q then p)”
- Note that a bi-conditional has the opposite truth values of the exclusive or

CSci 2011

Logical operators: Bi-conditional 2

- Let p = “You take this class” and q = “You get a grade”
- Then pq means “You take this class if and only if you get a grade”
- Alternatively, it means “If you take this class, then you get a grade and if you get a grade then you take (took) this class”

CSci 2011

Precedence of operators

- Just as in algebra, operators have precedence
- 4+3*2 = 4+(3*2), not (4+3)*2

- Precedence order (from highest to lowest):
¬ → ↔

- The first three are the most important

- This means that p q ¬r→s↔tyields: (p (q (¬r)) →s) ↔ (t)
- Not is always performed before any other operation

CSci 2011

Translating English Sentences

- Question 7 from Rosen, p. 17
- p = “It is below freezing”
- q = “It is snowing”

- It is below freezing and it is snowing
- It is below freezing but not snowing
- It is not below freezing and it is not snowing
- It is either snowing or below freezing (or both)
- If it is below freezing, it is also snowing
- It is either below freezing or it is snowing, but it is not snowing if it is below freezing
- That it is below freezing is necessary and
sufficient for it to be snowing

pq

p¬q

¬p¬q

pq

p→q

((pq)¬(pq))(p→¬q)

p↔q

CSci 2011

Translation Example 2

- Heard on the radio:
- A study showed that there was a correlation between the more children ate dinners with their families and lower rate of substance abuse by those children
- Announcer conclusions:
- If children eat more meals with their family, they will have lower substance abuse
- If they have a higher substance abuse rate, then they did not eat more meals with their family

CSci 2011

Translation Example 3

- “I have neither given nor received help on this exam”
- Let p = “I have given help on this exam”
- Let q = “I have received help on this exam”
- ¬p¬q

CSci 2011

Translation Example 4

- You can access the Internet from campus only if you are a computer science major or you are not a freshman.
- a (c f)
- You cannot ride the roller coaster if you are under 4 feet tall unless you are older than 16 years old.
- (f s) r
- r ( f s)

CSci 2011

Boolean Searches

(2011 OR 5471) AND yongdae AND “computer science”

- Note that Google requires you to capitalize Boolean operators
- Google defaults to AND; many others do not

CSci 2011

Bit Operations

- Boolean values can be represented as 1 (true) and 0 (false)
- A bit string is a series of Boolean values. Length of the string is the number of bits.
- 10110100 is eight Boolean values in one string

- We can then do operations on these Boolean strings
- Each column is its ownBoolean operation

01011010

10110100

11101110

CSci 2011

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