Let p, q, and r be the propositions
p :You have the flu. q :You miss the final examination.
r :You pass the course.
Express each of these propositions as an English sentence.
a) p q
If you have the fl
Section 5.2
The Pigeonhole Principle
1
Statement of the Principle
Theorem 1: If k is a positive integer and k+1 or more objects are placed into k boxes,
then there is at least one box containing two o
Section 4.1
Induction
The University of Texas at Dallas
1
www.utdallas.edu
Definition
Mathematical induction (over natural numbers) is a powerful
proof technique that is commonly used.
Given a state
Generalized Permutations and Combinations
Section 5.5
1
Permutations with Repetition
Theorem 1: The number of r permutations of a set of n objects with repetition allowed is
nr
Proof: This is a direct
Functions
Chapter 2
The University of Texas at Dallas
www.utdallas.edu
Definitions
The University of Texas at Dallas
www.utdallas.edu
continued
The University of Texas at Dallas
www.utdallas.edu
Examp
Sets
Chapter 2
The University of Texas at Dallas
www.utdallas.edu
Definition
The University of Texas at Dallas
www.utdallas.edu
Notation
The University of Texas at Dallas
www.utdallas.edu
Common Unive
Graphs and their
Properties
Jorge A. Cobb
The University of Texas at Dallas
1
Graphs
l
Graphs are a representation of relations,
highlighting a number of interesting problems
(graph theory)
l
l
isomor
Set Operations
Chapter 2
The University of Texas at Dallas
www.utdallas.edu
Cartesian
The University of Texas at Dallas
www.utdallas.edu
Example
The University of Texas at Dallas
www.utdallas.edu
Equa
The Binomial Coefficients
Section 5.4
1
Statement of the computation
Let x and y be variables and n a nonnegative integer. Then
n
n
j =0
(x + y ) = x n j y j
j
n
n n n n 1 n n 2 2
n nk k
n n 1 n
A 5:4" 5 5 Mt mum/m cal/W7?"
#
Z 0?wa ( dmt+$) Wm).
A L r: 5.1.1 1; may. .-+; and;
W.cfw_p:m- 5: cfw_0a, 04",0n
A. 6' S A U 4 m? is
a 4- S 4 ,3 gr4. Juarzg
W g 3rd cfw_ mural unload.
z I) I. ars
2+
Applications of the
Principle of InclusionExclusion
Part II
Jorge A. Cobb
The University of Texas at Dallas
1
Alternative form of
Inclusion-Exclusion
Let Ai be the subset of elements with property
Pi.
Logic
An Example of a
Boolean Algebra
The University of Texas at Dallas
Jorge Cobb
You know about algebra in math
(
2
)
, where
a+b
a and b, are variables whose values come from
some set: t
Equivalence Relations
Jorge Cobb
The University of Texas at Dallas
1
Review questions
What is the reflexive closure of the relation
cfw_(x, y): xy on the set of integers?
l Let R be the relation that
Chapter 3
Algorithms
Section 3.1
The University of Texas at Dallas
www.utdallas.edu
1
Algorithms
Definition: An algorithm is a finite set of precise instructions for
performing a computation or for s
Relations
Jorge Cobb
The University of Texas at Dallas
1
What are relations?
Relations are a formal means to specify
which elements from two or more sets are
related to each other
l Examples
l
l
l
l
l
Chapter 3
Growth of Functions
Section 3.2
The University of Texas at Dallas
www.utdallas.edu
1
How functions grow relative to each other
We will later analyze the complexity of an algorithm by countin
Linear Recurrence
Relations
Part I
Jorge Cobb
The University of Texas at Dallas
1
Solving recurrence relations
Setting up a recurrence relation is important
it corresponds to modeling a problem
l Sol
Chapter 3
Algorithm Complexity
Section 3.3
The University of Texas at Dallas
www.utdallas.edu
1
Algorithms Efficiency
To determine an algorithms efficiency, two
entities are measured:
The amount of
Proofs
Continuing on chapter 1
The University of Texas at Dallas
www.utdallas.edu
Definitions
A theorem is a valid logical assertion which
can be proved using:
axioms (statements which are given to
Graph Representation
Jorge A. Cobb
The University of Texas at Dallas
1
Representing graphs
l
Two major approaches
l
l
l
By listing the edges
As a matrix
The best approach depends on what we want
to do
Divide and Conquer
Algorithms
Part II
Jorge A. Cobb
The University of Texas at Dallas
Multiplication of integers
a = (an-1an-2.a0)2, b= (bn-1bn-2.b0)2
l Add the following partial products
l
l
For each
Partial Orders
Jorge A. Cobb
The University of Texas at Dallas
1
Partially ordered sets
Partial orders (relations that are reflexive,
antisymmetric, and transitive) can be used to
order the elements o
Shortest paths and
Planar graphs
Jorge A. Cobb
The University of Texas at Dallas
1
Weighted graphs
A weighted graph is a graph where numbers
(weights or costs) have been attached to
each edge.
l Examp
Connectivity
Jorge A. Cobb
The University of Texas at Dallas
1
Paths in graphs
A path of length n in an undirected graph is a
sequence of n edges such that successive
edges share a common endpoint.
l
Chapter 3
Algorithm Complexity
Section 3.3
The University of Texas at Dallas
www.utdallas.edu
1
Algorithms Efficiency
To determine an algorithms efficiency, two
entities are measured:
The amount of
5
The Processor:
Datapath
and Control
In a major matter,
no details are small.
French Proverb
5.1
Introduction 284
5.2
Logic Design Conventions 289
5.3
Building a Datapath 292
5.4
A Simple Implementat
EE/CE 4304: Computer Architecture
Bill Swartz
Dept. of EE
Univ. of Texas at Dallas
EEDG/CE4304 B. Swartz
1
Amdahls Law
2
History
Gene Amdahl
chief architect of IBM's first mainframe series
founder
Motivation
The Product Rule
The Sum Rule
The Subtraction Rule
Basics of Counting
Jorge A. Cobb
U. T. Dallas
October 18, 2017
Division Rule
Tree Diagrams
Motivation
The Product Rule
The Sum Rule
Table
Chapter 3
Growth of Functions
Section 3.2
The University of Texas at Dallas
www.utdallas.edu
1
Function growth relative to another function
We present the concept of a function g growing at least as