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
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 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
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
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
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
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
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
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
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
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
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
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
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
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
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
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 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
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,
CS70
Summer2016
DiscreteMathematicsandProbabilityTheory
geueueurillaSection3Solutions
DiscreteProbabilityIntro
1. Practice
a) Box1contains2redballsand1blueball.Box2contains3blueballsand1redball.A
coinistossed.Ifitfallsheadsup,box1isselectedandaballisdraw
CS 70
Discrete Mathematics and Probability Theory
Fall 2012
Vazirani
HW 1
Due Friday August 31
1. (5 pts.) Honor Code
Two CS 70 students Alice and Bob decide to work in a group. They collaborate to gure out
how to solve every question on the homework. The
CS 70
Discrete Mathematics and Probability Theory
Fall 2012
Vazirani
1
HW 2
Quantiers and predicates
Suppose P (n) is a predicate on the natural numbers, and suppose k N P (k ) P (k + 2).
For each of the following assertions below, state whether (A) it mu
CS 70
Discrete Mathematics and Probability Theory
Fall 2012
Vazirani
HW 2 Solutions
1 Quantiers and predicates
Suppose P (n) is a predicate on the natural numbers, and suppose k N P (k) P (k + 2).
For each of the following assertions below, state whether
CS 70
Discrete Mathematics and Probability Theory
Fall 2012
Vazirani
HW 3
1 Generalized DeMorgans Laws: An Exercise in Induction
Prove using induction
n
n
Ai c = (
Ai )c
i=1
i=1
Where Ac means A complement, the set of elements which lie in the space but w
CS 70
Discrete Mathematics and Probability Theory
Fall 2012
Vazirani
HW 3
1 Generalized DeMorgans Laws: An Exercise in Induction
Prove using induction
n
n
Ai c = (
Ai )c
i=1
i=1
Where Ac means A complement, the set of elements which lie in the space but w
CS 70
Discrete Mathematics and Probability Theory
Fall 2012
Vazirani
HW 4
Due Friday, September 21, 4:59pm
(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
Tab
CS 70
Discrete Mathematics and Probability Theory
Fall 2012
Vazirani
HW 5
Due Friday, September 28, 4:59pm
1
Lagrange Interpolation
This problem will have you practice with Lagrange interpolation. Here, we are looking for a
polynomial p(x) of degree at mo
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
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
Discrete Mathematics and Probability Theory
Summer 2016 Dinh, Psomas, and Ye
Guerilla Section 5 Sol
1. Propositional Logics
Which of the following propositions is true? In part e, Q(k) denotes the proposition "1 + 2 + + k =
k(k + 1)/2".
(a) (x N. x2
CS 70
Discrete Mathematics and Probability Theory
Summer 2016 Dinh, Psomas, and Ye
HW 7
Due Wednesday August 10 at 1:59 PM
1. Maximum Likelihood (14 points: 4/5/5)
Often, when you observe samples from a random distribution, you dont know the parameters wh