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 Theoretical Foundations of Computer Engg, Spring 08
Homework 2
Due: 7 Feb 2008, In class
Topics: Proofs, Propositions, Sets and Functions Reading: Sections 1.1, 1.2, 1.6, 1.7, 2.1, 2.2, 2.3
Problem 2.1
Use truth tables to show that the fol

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

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 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) —>

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 / 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 Theoretical Foundations of Computer Engg, Spring 08
Homework 3
Due: 14 Feb 2008, In class
Topics: Predicates, Proof by Induction Reading: Sections 1.3, 4.1
Problem 3.1
Let P (x) be the predicate "x = x2 ". If Z denotes the set of integers,

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

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 / 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 / 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 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 2.4
Lecture 12: More on functions; sequences
We have heard about injective, surjective, and bijective functions. Let us now use these ideas to
develop two important ways to transform functions; how to comput

CprE 310 / Fall 2016
Textbook coverage: Section 9.6
Lecture 24: Order relations
Partial and total orders
Recall our definition of a partial order relation: any relation R that is reflexive, anti-symmetric
and transitive is called a partial order relation

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 6.1
Lecture 17: Counting
Enumerating the elements of a set is central to several problems in computer engineering (and several
other technical disciplines.)
However, counting can be hard. Consider, for examp

CprE 310 / Fall 2016
Textbook coverage: Section 9.1
Lecture 22: Relations
Binary Relations
Having defined sets, functions, and so on, we now introduce relations. Relations capture pairwise
interactions between objects, and are a powerful generalization of

CprE 310 / Fall 2016
Textbook coverage: Section 9.3
Lecture 24: Relations (contd)
We have discussed some key properties of relations. We will now define a few more properties; but
keep in mind the three key ones: reflexivity, symmetry, and transititivity.

CprE 310 / Fall 2016
Textbook coverage: Section 1.7, 1.8
Lecture 6: Methods of proof
A good proof is like a well-written software program. The pieces that you need to put together may
not be evident right from the beginning, and takes practice and experie