Huan Long
Shanghai Jiao Tong University
Instructor
Huan Long ()
E-mail:
longhuan@sjtu.edu.cn
Office:
Dianxin Building, House
III,327
Office Hour:
13:00pm-15:00pm
every Monday
TA:
Jin Yang
Email:
jyjz2008(at)sjtu.edu.cn
Office:
Dianxin Building, Ho
Due: 2013/04/23 before class
Homework 8
If you prefer submitting your homework via email, please
send it to TA: jyjz2008@sjtu.edu.cn.
Problem 1. Show that if A is a transitive set, A+ is also a transitive set.
Problem 2. Show that A is a transitive set i
Due: 2013/04/28 before class
Homework 9
If you prefer submitting your homework via email, please
send it to TA: jyjz2008@sjtu.edu.cn.
Problem 1. Construct a one-to-one correspondence between the closed unit
interval [0, 1]
[0, 1] = cfw_x R | 0 x 1
and the
Due: 2013/05/07 before class
Homework 9
If you prefer submitting your homework via email, please
send it to TA: jyjz2008@sjtu.edu.cn.
Problem 1. Show for any sets X and Y , with Y containing at least two
elements, that there cannot be a one-to-one corresp
Q1. Find the value of
by Fixed point Iteration method correct to third decimal place.
3
x 3 15 0
15
root lies between
(2,3)
,
in
(2,3)
15
x
( x) ( x) 15 1
3
x
2x 2
By iteration method
Q2. Find all the roots of
f ( x) cos x x 2 x
,
,
,
,
.
xi 1 ( xi ) x0
There are at least four approaches available to investigate further a statistically significant omnibus chi-square test result. The first and easiest of the four procedures is calculating residuals.
A residual analysis identifies those specific cells maki
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Student book Task 1
Quantitative Analysis for Business
QAT1
Assignment 309.3.1-01 - Versio
Numerical Techniques
by Fixed point Iteration method correct to third decimal place.
Q1. Find the value of
3
x 3 15 0
15
root lies between
(2,3)
,
x
in
15
( x) ( x) 15 1
3
x
2x 2
By iteration method
Q2. Find all the roots of
f ( x) cos x x x
2
,
(2,3)
,
Due: 2013/04/15 before class
Homework 7
If you prefer submitting your homework via email, please
send it to TA: jyjz2008@sjtu.edu.cn.
Problem 1. Compare the following two statements, and give a proof of the
2nd item
1. =
cfw_A | A is inductive.
2. The cla
Due: 2013/04/08 before class
Homework 6
If you prefer submitting your homework via email, please
send it to TA: jyjz2008@sjtu.edu.cn.
Problem 1. What is the Russells Paradox? Explain its impact on the naive
set theory.
Problem 2. Show that there is no set
Axiomatic Set Theory
Huan Long
Shanghai Jiao Tong University
Naive set theory: any definable collection is a set.[Cantor 1870]
Is this enough?
Spring 2013
Paradox
Axiomatic System
ZFC
Spring 2013
Russell`s paradox(1902)
Bertrand Russell(1872-1970)
Britis
Two way of introducing new objects for
mathematical study
The axiomatic approach
Start with some primitive notions ,
A list of axioms dealing with the primitive notions.
The constructive approach
Some available objects,
Known properties of the known
CONSTRUCTION OF
THE REAL NUMBERS
Shanghai Jiao Tong University
Huan Long
Spring 2013
Set Theory
Motivation & technical roadmap
Continuous Object.
To show how mathematics can be embedded in set
theory.
Roadmap
Natural
Numbers
Integers
Rational
Numbers
R
Huan Long
Shanghai Jiao Tong University
Motivation
To discuss the size of sets. Given two sets A and B, we
want to consider such questions as:
Do A and B have the same size?
Does A have more elements than B?
Spring 2013
Equinumerosity
Equinumerosity
Due: 2013/03/04 before class
Homework 1
If you prefer submitting your homework via email, please
send it to TA: jyjz2008@sjtu.edu.cn.
Problem 1. List the members of the following set.
(a) cfw_x | x is the square of an integer and x < 100
(b) cfw_x | x is
Due: 2013/03/11 before class
Homework 2
If you prefer submitting your homework via email, please
send it to TA: jyjz2008@sjtu.edu.cn.
Problem 1. Show that if A B then
A
Problem 2. Show that for any set A,
P (A) = A.
B.
Problem 3. Show that for any sets A
Due: 2013/03/18 before class
Homework 3
If you prefer submitting your homework via email, please
send it to TA: jyjz2008@sjtu.edu.cn.
Problem 1. Suppose that we attempted to generalize the Kuratowski denitions of ordered pairs to the ordered triples by de
Due: 2013/03/25 before class
Homework 4
If you prefer submitting your homework via email, please
send it to TA: jyjz2008@sjtu.edu.cn.
Problem 1. R X Y , A, B are sets, show that
a)
R [ A]
cfw_R[B ] | B A, A =
b)
R[A] R[B ] R[A B ]
c)
If R is an injectiv
Due: 2013/04/01 before class
Homework 5
If you prefer submitting your homework via email, please
send it to TA: jyjz2008@sjtu.edu.cn.
Problem 1.
1. Show that R is symmetric i R1 R.
2. Show that R is transitive i R R R.
Problem 2. Show that R is a symmetri