Articles and Excerpts
Volume 1
Art of Problem Solving
2005, 2006, AoPS Incorporated.
This book may be freely reproduced in print or electronic form for personal
or classroom use, provided that any reproduction includes a copy of this
page.
Published by:
Math 108, problem set 03
Outline due: Wed Feb 12
Completed version due: Mon Feb 17
Last revision due: Mon Mar 17
Exercises (to be done but not turned in): 8.3-8.5, 8.8-8.10; 9.2-9.5, 9.8-9.9.
Problems to be turned in: All numbers refer to problems in the
Floating point
IEEE Floating Point Standard
Rounding
Floating Point Operations
Today
Mathematical properties
IEEE Floating point
Floating point representations
Encodes rational numbers of the form V=x*(2 y)
Useful for very large numbers or numbers close
Integers
Numeric Encodings
Programming Implications
Basic operations
Today
Programming Implications
Integers in C
C supports several integral data types
Note the unsigned modifier
Also note the asymmetric ranges
C data type(32b)
Size
Minimum
Maximum
cha
Bits and Bytes
Why bits?
Binary/hexadecimal
Today
Byte representations
Boolean algebra
Expressing in C
Why dont computers use Base 10?
Base 10 number representation
Digit in many languages also refers to fingers/toes
Of course, decimal (from Latin
CHAPTER
8
Answers and Hints
These pages contain answers to every exercise appearing in the book. Before
getting overly excited, though, keep in mind that a thorough understanding of
an exercise encompasses much more than just the correct answer. It will i
CHAPTER
1
Logical Foundations
1.1
Statements and Open Sentences
Certain words and phrases are ubiquitous in mathematical discourse because
they convey the logical framework of the ideas being presented. For this reason
they arise naturally in everyday lan
CHAPTER
3
Proof Techniques
3.1
A Case for Proof
Towards the end of the rst chapter we met Bertrands Postulate, which states
that for all n 2 there is a prime between n and 2n. We veried this claim for
values of n from n = 2 through n = 10. One could easil
CHAPTER
2
Set Theory
2.1
Presenting Sets
Certain notions which we all take for granted are harder to dene precisely
than one might expect. In Taming the Innite: The Story of Mathematics, Ian
Stewart describes the situation in this way:
The meaning of numb
Math 108, problem set 02
Outline due: Wed Feb 05
Completed version due: Mon Feb 10
Last revision due: Mon Mar 17
Exercises (to be done but not turned in): 6.2-6.5, 6.7-6.8, 6.10, 6.12-6.14; 7.3, 7.5.
(Corrected Mon Feb 03.)
Problems to be turned in: All n
Math 108, problem set 01b
Due: Mon Feb 03
Last revision due: Mon Feb 17
Problems to be turned in:
The goal of this problem set is to outline the proof of the following theorem.
Theorem 1. Let n be a positive integer. If v1 , v2 , v3 , v4 , v5 are containe
Math 108, problem set 05
Outline due: Wed Mar 05
Completed version due: Mon Mar 10
Last revision due: Wed Apr 23
Exercises (to be done but not turned in): 12.1, 12.4, 12.5, 12.6, 12.8, 12.9.
Problems to be turned in: All numbers refer to problems in the Y
Math 108, problem set 07
Outline due: Wed Apr 02
Completed version due: Mon Apr 07
Last revision due: Wed Apr 23
Exercises (to be done but not turned in): 16.3, 16.5, 16.9, 17.1, 17.6.
Problems to be turned in: All numbers refer to problems in the Yellow
Sample Exam 2
Math 108, Spring 2014
1. (14 points) Let X be a set, and let A be a collection of subsets of X (i.e., each S A is
a subset of X ). Dene what it means for A to be a partition of X .
In questions 24, you are given a statement. If the statement
Writing proofs
Tim Hsu, San Jos State University
e
Revised January 2014
Contents
I
Fundamentals
5
1 Denitions and theorems
5
2 What is a proof ?
5
3 A word about denitions
6
II
The structure of proofs
8
4 Assumptions and conclusions
8
5 The if-then method
Sample Exam 1
Math 108, Spring 2014
1. (12 points) Let X and Y be sets.
(a) Dene what it means for R to be a relation from X to Y .
(b) Dene what it means for R to be a relation on X .
2. (14 points) As usual, knights always tell the truth, knaves always
Math 108, problem set 04
REVISED FRI MAR 14
Outline due: Wed Feb 26
Completed version due: Mon Mar 03
Last revision due: WED APR 23
Denitions:
block If cfw_A | I is a partition of a set X , we call the A the blocks of the partition.
Exercises (to be done
Format and topics
Exam 2, Math 108
General information. Exam 2 will be a timed test of 75 minutes, covering Chapters 1012
and 1417 of the Yellow Book and Parts IIII (Sections 119) of the proof notes. No books, notes,
calculators, etc., are allowed. Most o
Math 108, problem set 01a
Due: Wed Jan 29
Last revision due: Mon Feb 17
Problems to be done but not turned in: (Smullyan) Ch. 3, problems 2638; Ch.
8, problems 109126.
Problems to be turned in:
Problems 14: For each of the theorems listed in these problem
Math 108, problem set 06
Outline due: Wed Mar 12
Completed version due: Mon Mar 17
Last revision due: Wed Apr 23
Denitions: Let f : X Y be a function, let A be a subset of X , and let B be a
subset of Y .
restriction We dene the restriction of f to A to b