Partial Solutions and Comments, Math 74 HW2 and HW3
General Comments: Please split up your proofs into paragraphs when appropriate, such as in longer proofs like HW3, Problem 9. Also, you should dene most ob jects that youre using, though you dont need to
1 Distance
J MUSCAT
1
Metric Spaces
Dr J Muscat 2003
1
Distance
Metric spaces can be thought of as very basic spaces having a geometry, with only a few axioms. They are generalizations of the real line, in which some of the theorems that hold for R remain
MATH 74, SPRING 2009
DAN BERWICK EVANS
1. Introduction: Linguistic and Cultural Gaps in Mathematics Why dont middle and lower division classes adequately prepare students for upper division mathematics? This question has troubled teachers and students ali
MATH 74 HOMEWORK 10
Due Wednesday May 6th at 3:10pm. (1) Prove that homotopy denes an equivalence relation on the set of based loops in a topological space. (2) Prove that multiplication in the fundamental group is well-dened. That is, given equivalence c
MATH 74 HOMEWORK 9
Due Wednesday April 24th at 3:10pm. Throughout, let G be a group. Many of the following problems are from (or adapted from) Heresteins Topics in Algebra. (1) Let Z (G) be the center of G and suppose that G/Z (G) is cyclic. Prove that G
MATH 74 HOMEWORK 8
Due Wednesday April 17th at 3:10pm. Throughout, let G be a group. Many of the following problems are from (or adapted from) Heresteins Topics in Algebra. (1) Let (n) be the Euler function, so (n) is the number of positive integers less
MATH 74 HOMEWORK 7
If you dont like your analyst, see your local algebraist! Gert Almkvist Due Wednesday April 8th at 3:10pm. Throughout, let G be a group. Many of the following problems are from (or adapted from) Heresteins Topics in Algebra. (1) In the
MATH 74 HOMEWORK 6
Due Friday March 13th at 3:10pm. Throughout, let X and Y be metric spaces. (1) Let A and B be sets and f a function. Show that f 1 (A B ) = f 1 A f 1 B and that f 1 (A B ) = f 1 A f 1 B . (2) In a general metric space, do open and close
MATH 74 HOMEWORK 5
Due Friday March 8th at 3:10pm. Throughout, let X and Y be metric spaces. (1) Let cfw_xn be a Cauchy sequence, and suppose that a subsequence cfw_xni converges. Show that n=0 i=0 the whole sequence cfw_xn must converge. (2) Show that
MATH 74 HOMEWORK 4
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Due Wednesday February 25th at 3:10pm. Throughout, let X and Y be metric spaces. (1) Let x X . Prove that cfw_x is a closed set, and that the sets X and are both open and closed in X . (2) Show that the constant functions f : X Y , x
Solution Sketches and Comments, Math 74 HW4
The set is open is not equivalent to The set is not closed, and similarly, The set is closed is not equivalent to The set is not open. You actually showed this in Problem 4.1 by proving that a set can be both cl
MATH 74 HOMEWORK 3
To see a world in a Grain of Sand And a Heaven in a Wild Flower Hold Innity in the palm of your hand And Eternity in an hour. . . -William Blake Due Wednesday February 18th at 3:10pm. (1) Prove that |2X | = 2|X | where 2X is the power s
MATH 74 HOMEWORK 2
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Due Wednesday February 4th at 3:10pm. (1) (a) Dene the statement f : R R is continuous at a R. (b) Write the negation of the statement in (a), without using not. (c) Choose an example of a function that is not continuous at a point an
MATH 74 HOMEWORK 1
DAN BERWICK EVANS
The following is due Wednesday, January 28th at 3:10 pm. (1) (Eccles 1.2) Construct truth tables for the statements: (a) not (P and Q) (b) (not P ) or (not Q) (c) P and (not Q) (d) (not P ) or Q. (2) Convert the follow
Partial Solutions and Comments, Math 74 HW2 and HW3
General Comments: Please split up your proofs into paragraphs when appropriate, such as in longer proofs like HW3, Problem 9. Also, you should dene most ob jects that youre using, though you dont need to
MATH 74 FINAL REVIEW
(1) Basic Set Theory, Logic and Functions: set operations (intersection, complement, power set, Cartesian product, etc.), predicate logic, truth tables, basic properties of functions (injectivity, composition, inverse image, etc.) (2)
Objects distant and near from Interactive Mathematics Miscellany and Puzzles
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Books for Strong Students
Books, Classes and Forum Designed for High-Performing Math Students www.artofproblemsolving.com
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Attention: A once a year event! The
Common Errors in Writing Mathematics The following things are often confused: equations, expressions, functions.
An equation is a statement that two things are equal, like x3 + 3 x 2 = 0 x+1 or (x + 1)(y 2) or y = x2 .
An expression is some mathematical s
A Mathematicians Apology
G. H. Hardy
First Published November 1940 As fifty or more years have passed since the death of the author, this book is now in the public domain in the Dominion of Canada. First Electronic Edition, Version 1.0 March 2005 Publishe
A Mathematical Writing Checklist
Below is a list of guidelines you should follow for mathematical papers. For more details, please consult A Guide to Writing Mathematics . Is your pap er neatly typ ed? If you write the equations by hand, make sure that yo
A Guide to Writing Mathematics
Dr. Kevin P. Lee
Intro duction
This is a math class! Why are we writing?
There is a go o d chance that you have never written a paper in a math class before. So you might be wondering why writing is required in your math cla
Metric space - Wikipedia, the free encyclopedia
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Metric space
From Wikipedia, the free encyclopedia
In mathematics, a metric space is a set where a notion of distance (called a metric) between elements of the set is defined. The metric space which
MATH 74 SYLLABUS, SPRING 2009
DAN BERWICK EVANS
Contact Information email: devans@math.berkeley.edu. Oce: 775 Evans. Oce Hours: Monday 4-5, Thursday 1:30-2:30 or by appointment. Course Website: http:/math.berkeley.edu/~devans. There are links to numerous
Surprises from Mathematics Education Research: Student (Mis)use of Mathematical Denitions
Barbara S. Edwards and Michael B. Ward
1. INTRODUCTION. The authors of this paper met at a summer institute sponsored by the Oregon Collaborative for Excellence in t
George M. Bergman
Spring 2006
Supplementary material
Some notes on sets, logic, and mathematical language
These are generic notes, for use in Math 110, 113, 104 or 185. T h e s e p a g e s d o n o t d e v e l o p i n d e t a i l t h e d e n i t i o n s a
MATH 74 QUIZ 4
Pick one of the following questions and make a 10-15 minute appointment to present it to me before Wednesday May 13th. You will also be responsible for any denitions or theorems you use in your proof or explanation, though I will only ask f
MATH 74 QUIZ 3 REVIEW
The following are the major topics: (1) Groups and Homomorphisms: Denitions, uniqueness of inverses, elementary properties of homomorphisms (e.g. (x1 ) = (x)1 ) with proofs, kernels of homomorphisms, abelian groups, order of a group.