CprE 310 Theoretical Foundations of Computer Engg, Spring 08
Homework 1
Due: 31 Jan 2008, In class
Problem 1.1
Prove the following statement. For any natural number k 1, if 3k + 7 is odd, then k must be even.
Problem 1.2
We know (from class) that

CprE 310
Exam #2
11/5/2015
Name: C H K 1 S C H U Score: /100
1. (Total / 8 points) Let P(n) be the statement that (9" — 5") is divisible by 4. This question proves
that P(n) is true for all n 2 0.
(a) (2 points) Complete the basis step by proving that P

Lecture 6 Outline
Ch. 2.3 Functions
Definition of a Function
Domain, Codomain
Image, Preimage
Range
Injection, Surjection, Bijection
Inverse Function
Function Composition
Graphing Functions
Floor, Ceiling, Factorial
CprE 310
Lec 6
1
Functions
De

Spring 2016
CprE 310
Homework #10 Solutions
4/14/2016
Chapter 9.1 (Starting at p. 581)
1. (8 points)
Exercise 4(a). Being taller than is not reflexive (I am not taller than myself), nor symmetric (I am taller than my
daughter, but she is not taller

Fall 2015
CprE 310
Homework #9 Solutions
11/12/2015
Chapter 7.3 (Starting at p. 475)
1. (6 points) Exercise 2. ( | ) = ( / () = (5/8)(2/3)/(3/4) = 5/9.
2. (10 points) Exercise 6. Let be the event that a randomly chosen soccer player uses st

Fall 2015
CprE 310
Homework #10 Solutions
11/19/2015
Chapter 9.1 (Starting at p. 581)
1. (8 points)
Exercise 4(a). Being taller than is not reflexive (I am not taller than myself), nor symmetric (I am taller than my
daughter, but she is not

Fall 2015
CprE 310
Homework #12 Solutions
12/10/2015
Chapter 10.5 (Starting at p. 703)
1. (4 points) Exercise 4. This graph has no Euler circuit, since the degrees of vertex f and vertex c are odd. There
is an Euler path between the two vertices of

CprE 310 / Fall 2016
Textbook coverage: Section 5.2
Lecture 16: Induction (II)
While proving a statement by induction, it is often fruitful to ask yourself the following questions:
Is the domain of discourse the nonnegative integers (or some subset of th

CprE 310 / Fall 2016
Textbook coverage: Section 6.2, 6.4, 6.5
Lecture 20: Counting (contd)
Counting in two ways
Combinatorial proofs usually involve counting the elements of some set in two different ways, and
equating the results. This notion of counting

CprE 310 K ET ‘
Exam #2
4/5/2016
Name: C H K I S C H U Score: /100
1. (Total /8 points) Let P(n) be the statement 1 - 1! + 2- 21+ 3 - 3! + + n - n! = (n + 1)! — 1.
(a) (2 points) Prove that P(l) is true.
l-l! :1 = OHM-1
(b) (5 points) Prove that P(k) —>

Ch. 1 The Foundations: Logic and Proofs
Propositional Logic
1.1 The Language of Propositions
1.2 Applications
1.3 Logical Equivalences
Lec. 1
Lec. 2
Predicate Logic
1.4 The Language of Quantifiers
1.4 Logical Equivalences
1.5 Nested Quantifiers
Le

Ch. 1 The Foundations: Logic and Proofs
Propositional Logic
1.1 The Language of Propositions
1.2 Applications
1.3 Logical Equivalences
Lec. 1
Lec. 2
Predicate Logic
1.4 The Language of Quantifiers
1.4 Logical Equivalences
1.5 Nested Quantifiers
Le

Ch. 1 The Foundations: Logic and Proofs
Propositional Logic
1.1 The Language of Propositions
1.2 Applications
1.3 Logical Equivalences
Lec. 1
Lec. 2
Predicate Logic
1.4 The Language of Quantifiers
1.4 Logical Equivalences
1.5 Nested Quantifiers
Le

Ch. 1 The Foundations: Logic and Proofs
Propositional Logic
1.1 The Language of Propositions
1.2 Applications
1.3 Logical Equivalences
Lec. 1
Lec. 2
Predicate Logic
1.4 The Language of Quantifiers
1.4 Logical Equivalences
1.5 Nested Quantifiers
Le

CprE 310 / Fall 2016
Textbook coverage: Section 2.5
Lecture 14: Summations, Sizes of Infinite Sets
We introduced summations in the previous lecture. Recall that a summation of a sequence is itself a
sequence; summations can be computed using the forward o

CprE 310 / Fall 2016
Textbook coverage: Section 2.5
Lecture 13: Recursion and Summations
We have defined what recursion is, and seen how to model sequences using recursive constructions.
Now we will ask the reverse question; given a recursion, how can we

CprE 310 / Fall 2016
Textbook coverage: Section 2.3
Lecture 11: Types of Functions
In the last lecture, we gave a definition of a function, and established various notions such as its
domain, co-domain, and range. We also encountered ideas such as the ima

CprE 310 / Fall 2016
Textbook coverage: Section 9.4
Lecture 26: Introduction to Graph Theory
We are transitioning into the last topic of the course (and one of the more important ones) and that
is graph theory. But before we discuss graphs one last topic

CprE 310 / Fall 2016
Textbook coverage: Section 10.2
Lecture 27: Graph Theory
Let us continue our discussion on (undirected) graphs, and develop some basic theory that will let us
solve some interesting problems. But first, some history.
Paths, etc.
Let u

CprE 310 / Fall 2016
Textbook coverage: Section 6 (all)
Lecture 21: Counting (fin)
Counting with recurrences
The Tower of Hanoi problem is a famous problem in combinatorics. Typically one introduces the
problem using a picture of 3 pegs with a stack of di

CprE 310 / Spring 2017
Practice Problem Set 2
1. Recall that a number is called rational if it can be written as the ratio of two integers.
Numbers that are not rational are called irrational. Prove by contraposition the following statement:
If r is irra

CprE 310 /Spring 2017
Practice Problems set 3
1. Prove the following distributive law for sets A, B, C:
A (B C) = (A B) (A C)
You can use any method you like. For example, you could consider an element x A(B C)
and construct a chain of logical deductions

CprE 310: Theoretical Foundations of Computer Engineering
Due March 11, 2008
Friendly World or Not? 1 Goal
The goal of this assignment is to provide an understanding of how large graphs are represented in a computer, and experience in designing and

CprE 310 Theoretical Foundations of Computer Engg, Spring 08
Homework 4
Due: 21 Feb 2008, in class
Topics: Recursive Definitions, Proofs Reading: Sections 4.3, 4.4
Problem 4.1
Prove that for any n 6, an equilateral triangle can be partitioned into