CS 70
Discrete Mathematics and Probability Theory
Fall 2012
Vazirani
HW 4
Due Friday, September 21, 4:59pm
1. (4+6 pts.) Stable Marriage
Consider the following instance of the stable marriage problem:
Man
1
2
3
highestlowest
B
C
A
A
B
C
Woman
C
A
B
A
B
C
CS 70
Discrete Mathematics and Probability Theory
Fall 2012
Vazirani
HW 6
1 Interpolation practice
Find a polynomial h(x) = ax2 + bx + c of degree at most 2 such that h(0) 3 (mod 7), h(1) 6
(mod 7), and h(2) 6 (mod 7) with Lagrange Polynomials.
How many d
CS 70
Discrete Mathematics and Probability Theory
Spring 2016
Rao and Walrand
HW 4
Due Thursday 18th at 10PM
1. Amaze your friends!
(a) You want to trick your friends into thinking you can perform mental arithmetic with very large
numbers What are the las
CS 70
Spring 2016
Discrete Mathematics and Probability Theory
Rao and Walrand
HW 3
Due Thursday February 11th at 10PM
1. Homework process and study group Who else did you work with on this homework? List names
and student IDs. (In case of hw party, you ca
CS 70
Discrete Mathematics and Probability Theory
Spring 2016
Rao and Walrand
HW 2
Due Thursday February 4th at 10PM
1. (5 points)
Use induction to prove that for all positive integers n, all of the entries in the matrix
n
1 0
3 1
are 3n.
2. (5 points) Di
CS 70
Discrete Mathematics and Probability Theory
Fall 2012
Vazirani
HW 5
Due Friday, September 28, 4:59pm
1 Lagrange Interpolation
[Taken From: Summer 2012 Homework 3, but with dierent values, degree 3 instead of 2, and
mod 11 instead of mod 7.]
This pro
1
T/F
Last updated 2016.07.29 00:51
1
T/F
(3 points each) Circle T for True or F for False. We will only grade the answers, and are unlikely to
even look at any justifications or explanations.
(a)
(b)
(c)
(d)
(e)
T
F (P (Q R) (P Q) (P R). (T) LHS P (Q R)
CS 70
Fall 2015
Discrete Mathematics and Probability Theory
Note 0
Review of Sets and Mathematical Notation
A set is a well dened collection of objects. These objects are called elements or members of the set, and
they can be anything, including numbers,
CS 70
Discrete Mathematics and Probability Theory
Fall 2015
1
Note 4
The Stable Marriage Problem
In the previous note, we discussed the powerful proof technique of induction. In this note, we apply induction to analyze the solution to an important problem
CS 70
Discrete Mathematics and Probability Theory
Fall 2015
1
Note 3
Mathematical Induction
Introduction. In this note, we introduce the proof technique of mathematical induction. Induction is a
powerful tool which is used to establish that a statement ho
CS 70
Fall 2015
1
Discrete Mathematics and Probability Theory
Note 2
Proofs
In science, evidence is accumulated through experiments to assert the validity of a statement. Mathematics,
in contrast, aims for a more absolute level of certainty. A mathematica
Practice Test (for midterm 1)
Spring 2017
1) In a study of blood pressure and number of children, it turns out that there is a strong positive
correlation between blood pressure and how many children they have. True or False:
a. People with high blood pre
HW04 - Correlation and Regression Method
Stat 20 & 131A, Spring 2017, Prof. Sanchez
Due Feb-16
1) A class of 15 students happens to include 5 basketball players. True or False, and explain: the
relationship between heights and weights for this class shoul
CS70
Summer2016
DiscreteMathematicsandProbabilityTheory
geueueurillaSection3Solutions
DiscreteProbabilityIntro
1. Practice
a) Box1contains2redballsand1blueball.Box2contains3blueballsand1redball.A
coinistossed.Ifitfallsheadsup,box1isselectedandaballisdraw
CS 70
Fall 2015
1
Discrete Mathematics and Probability Theory
Note 1
A Brief Introduction
Welcome to Discrete Math and Probability Theory! You might be wondering what youve gotten yourself
into were delighted to tell you that the answer is something quite
CS 70
Spring 2016
1
Discrete Mathematics and Probability Theory
Rao and Walrand
Note 6
Modular Arithmetic
Suppose you go to bed at 23:00 oclock and want to get 8 hours of sleep. What time should you set your
alarm for? Clearly, the answer is not (23 + 8 =
CS 70
Spring 2016
1
Discrete Mathematics and Probability Theory
Rao and Walrand
Note 7
Bijections
The notion of a mathematical function, i.e. a mapping f from an input set A to an output set B, is ubiquitous
in our everyday lives. For example, your profes
CS 70
Discrete Mathematics and Probability Theory
Spring 2016
Rao and Walrand
Discussion 3B
1. Tournament
A tournament is dened to be a directed graph such that for every pair of distinct nodes v and w, exactly one
of (v, w) and (w, v) is an edge (represe
CS 70
Spring 2016
Discrete Mathematics and Probability Theory
Rao and Walrand
Discussion 2B Sol
1. Stable Marriage
Consider the following list of preferences:
Men
A
B
C
D
Preferences
4>2>1>3
2>4>3>1
4>3>1>2
3>1>4>2
Women
1
2
3
4
Preferences
A>D>B>C
D>C >A
CS 70
Discrete Mathematics and Probability Theory
Spring 2016
Rao and Walrand
HW 5
Due Thursday 18th at 10PM
1. Proof practice
The purpose of this problem is to practice formally proving a statement, when you intuitivelly "know"
why its true.
Suppose that
CS 70
Spring 2016
Discrete Mathematics and Probability Theory
Rao and Walrand
HW 1
Due Thursday January 28 at 10PM
1. (3 points) Wasons experiment:2
Suppose we have four cards on a table:
1st about Alice, 2nd about Bob, 3rd about Charlie, and 4th about D
CS 70
Discrete Mathematics and Probability Theory
Spring 2016
Rao and Walrand
Note 11
Innity and Countability
Cardinality
How can we determine whether two sets have the same cardinality (or size)? The answer to this question,
reassuringly, lies in early g
CS 70
Fall 2015
1
Discrete Mathematics and Probability Theory
Note 5
Graph Theory: An Introduction
One of the fundamental ideas in computer science is the notion of abstraction: capturing the essence or the
core of some complex situation by a simple model
CS 70
Spring 2016
Discrete Mathematics and Probability Theory
Rao and Walrand
Note 9
Error Correcting Codes
We will consider two situations in which we wish to transmit information on an unreliable channel. The
first is exemplified by the internet, where
CS 70
Spring 2016
Discrete Mathematics and Probability Theory
Rao and Walrand
Note 8
Polynomials
Polynomials constitute a rich class of functions which are both easy to describe and widely applicable in
topics ranging from Fourier analysis to computationa
Introduction
Lecture 6: Commitment Devices
ECON 119
UC Berkeley
Grant Graziani
June 27 2017
Introduction
Introduction
Announcements
1. Problem set 1 is due July 6 at 4pm.
Outline
1. Commitment devices
2. Evidence for demand for commitment
3. Welfare conse
GEOGRAPHY 20: GLOBALIZATION
Spring 2017
Tues/Thurs, 3:30-5:00 PM
100 Lewis Hall
Instructor:
GSIs:
Dr. John Stehlin ([email protected])
Nicholas Anderman ([email protected])
Rebecca Coates-Maldoon ([email protected])
Bridget Martin (martb2
Introduction
Lecture 1: Introduction
ECON 119
UC Berkeley
Grant Graziani
June 19 2017
Introduction
Introduction
Announcements
1. Take a look at the syllabus.
2. Dont forget to purchase an iClicker from the bookstore.
3. Check out the bCourses site for fut
2
1
TO CATCH A MAGIKARP
T/F
(4 points each) Circle T for True or F for False. We will only grade the answers, and are unlikely to
even look at any justifications or explanations.
2
(a)
Given some sample space = cfw_1, 2, 3, and events A = cfw_1, 2 and B =