General Tips on Direct Proofs and Proof by Counterexample
Proof Methods: If you are asked to prove an assertion of the form:
If Hypotheses then Conclusion
You may use any of the following methods:
1. Direct Proof. This is the most straightforward. You (1)
CHAPTER 6
Counting Methods and the Pigeonhole Principle
Section 6.1 Basic Properties
Multiplication Principle. If an activity can be constructed in successive steps and step 1 can be
done in
ways, step 2 can be done in
ways, ., and step can be done in way
CHAPTER 5
Introduction to Number Theory
Section 5.1 Divisors
Definition. Let and be integers,
. We say that divides if there exists a unique
integer satisfying
. We call the quotient and a divisor or factor of . If divides ,
we write
. If does not divide
CHAPTER 4
Algorithms
Section 4.1 Introduction
Definition. An algorithm is a finite set of precise instructions for performing a computation or
for solving a problem.
Remark. The word algorithm is derived from Al-Khowrizm, the last name of the 9th century
CHAPTER 3
Functions, Sequences, and Relations
Section 3.1 Functions
Definition. Let and be two sets. A function
peoduct
having the property that for each
We sometimes denote a function from to as
from to is a subset of the Cartesian
there is exactly
with
CHAPTER 2
Proofs
Section 2.1 Direct Proofs and Counterexamples
Proof Methods: If you are asked to prove an assertion of the form:
If Hypothesis then Conclusion
You may use any of the following methods:
1. Direct Proof. This is the most straightforward. Yo
CHAPTER 1
Sets and Logic
Section 1.1 Sets
Definition. A set is a collection of objects. The objects in a set are also called the elements or
members, of the set.
Notation. If is an element of a set , then we write
. Otherwise, we write
.
Definition. Let b
MATH-223 Discrete Mathematics I
Test III Study Guide
Disclaimer
This document is provided as a study guide only. It does not represent what will be on
the Test. It highlights the important topics that you must know for the Test. The Test
may include quest
MATH-223 Discrete Mathematics I
Test II Study Guide
Disclaimer
This document is provided as a study guide only. It does not represent what will be on
the Test. It highlights the important topics that you must know for the Test. The Test
may include questi
MATH-223 Discrete Mathematics I
Test I Study Guide
Disclaimer
This document is provided as a study guide only. It does not represent what will be on
the Test. It highlights the important topics that you must know for the Test. The Test
may include questio