Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I:
Homework Assignment 5
Assigned Tuesday October 8; due Friday October 18, 2PM at SS 1071
web version: http:/www.math.toronto.edu/drorbn/classes/
MAT157Y
PROBLEM SET 1
DUE IN CLASS ON MONDAY, SEPTEMBER 19
The symbols a, b, x, y all represent real numbers.
1. Use the properties of numbers listed in class to prove the following:
(i) If a > 0 and
Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I:
Math 157 Analysis I Solution of Term Exam 3
web version:
http:/www.math.toronto.edu/drorbn/classes/0203/157AnalysisI/TermExam3/Solution.html
Pr
Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I:
Homework Assignment 21
Assigned Tuesday March 4; due Friday March 14, 2PM at SS 1071
web version: http:/www.math.toronto.edu/drorbn/classes/020
Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I:
Math 157 Analysis I Solution of Term Exam 1
web version:
http:/www.math.toronto.edu/drorbn/classes/0203/157AnalysisI/TermExam1/Solution.html
Pr
Chapter 6
1. For which of the following functions
f is there a continuous function F with domain
such that F ( x) = f ( x) for all x in the domain of f ?
x2 4
, Yes. There is such a function g ( x )
Math 157 Analysis I Term Exam 1
University of Toronto, October 21, 2002.
Student ID:
Name:
Solve the following 5 problems.
Each is worth 20 points although they may have
unequal diculty. Write your an
Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I:
Homework Assignment 22
Assigned Tuesday March 11; due Friday March 21, 2PM at SS 1071
web version: http:/www.math.toronto.edu/drorbn/classes/02
Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I:
Homework Assignment 8
Assigned Tuesday October 29; due Friday November 8, 2PM at SS 1071
web version: http:/www.math.toronto.edu/drorbn/classes
Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I:
Homework Assignment 11
Assigned Tuesday November 19; due Friday November 29, 2PM at SS 1071
web version:
http:/www.math.toronto.edu/drorbn/clas
UNIVERSITY OF TORONTO
Faculty of Arts and Sciences
APRIL/MAY EXAMINATIONS 2003
Math 157Y Analysis I Final Exam
Dror Bar-Natan
April 28, 2003
Student ID:
Name:
Solve the following 6 problems.
Each is w
Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I:
Homework Assignment 20
Assigned Tuesday February 24; due Friday March 7, 2PM at SS 1071
web version: http:/www.math.toronto.edu/drorbn/classes/
Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I:
Homework Assignment 14
Assigned Tuesday January 7; due Friday January 17, 2PM at SS 1071
web version: http:/www.math.toronto.edu/drorbn/classes
Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I:
Homework Assignment 19
Assigned Tuesday February 4; due Monday February 24 at the tutorials
web version: http:/www.math.toronto.edu/drorbn/clas
Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I:
Homework Assignment 24
Assigned Tuesday March 25; due Friday April 4, 2PM at SS 1071
web version: http:/www.math.toronto.edu/drorbn/classes/020
Math 157 Analysis I Term Exam 2
University of Toronto, December 2, 2002.
Student ID:
Name:
Solve the following 5 problems.
Each is worth 20 points although they may have
unequal diculty. Write your an
Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I:
Homework Assignment 25
Assigned Tuesday April 1; due Friday April 11, 2PM at SS 1071
web version: http:/www.math.toronto.edu/drorbn/classes/020
Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I:
Homework Assignment 16
Assigned Tuesday January 21; due Friday January 31, 2PM at SS 1071
web version: http:/www.math.toronto.edu/drorbn/classe
Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I:
Homework Assignment 21
Assigned Tuesday March 4; due Friday March 14, 2PM at SS 1071
web version: http:/www.math.toronto.edu/drorbn/classes/020
Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I:
Homework Assignment 20
Assigned Tuesday February 24; due Friday March 7, 2PM at SS 1071
web version: http:/www.math.toronto.edu/drorbn/classes/
Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I:
Homework Assignment 17
Assigned Tuesday January 28; due Friday February 7, 2PM at SS 1071
web version: http:/www.math.toronto.edu/drorbn/classe
MAT157 - Proofs Tutorial
1
Easier examples
1.1
Direct Proof - Construction
There are two situations where proof by construction is a reasonable approach. First, are existence statements.
Any statement
Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I:
Homework Assignment 3
Assigned Tuesday September 24; due Friday October 4, 2PM at SS 1071
web version: http:/www.math.toronto.edu/drorbn/classe
Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I:
Homework Assignment 4
Assigned Tuesday October 1; due Friday October 11, 2PM at SS 1071
web version: http:/www.math.toronto.edu/drorbn/classes/
Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I:
Homework Assignment 1
assigned Sep. 10; due Sep. 24, 2PM at SS 1071
Required reading
All of Spivak Chapter 1.
To be handed in
From Spivak Chapt
Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I:
Math 157 Analysis I Solution of Term Exam 4
web version:
http:/www.math.toronto.edu/drorbn/classes/0203/157AnalysisI/TermExam4/Solution.html
Pr
Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I:
Math 157 Analysis I Solution of the Final Exam
web version: http:/www.math.toronto.edu/drorbn/classes/0203/157AnalysisI/Final/Solution.html
Pro
Chapter 4
3. Draw the set of all point (x,y) satisfying the following conditions.
ii) x + a > y + b
iv) y x
2
vi) x + y < 1
1
is an integer.
x+ y
viii)
x) x < y < x
2
4
Notes: the border is not i