Huan Long
Shanghai Jiao Tong University
Instructor
Huan Long ()
E-mail:
[email protected]
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
Ans 1(a):
A set is a group or collection of objects or numbers, considered as an entity unto itself.
Sets are usually symbolized by uppercase, italicized, boldface letters such as A, B, S, or Z.
Each object or number in a set is called a member or element
Ans 1(a):
A set is a group or collection of objects or numbers, considered as an entity unto itself.
Sets are usually symbolized by uppercase, italicized, boldface letters such as A, B, S, or Z.
Each object or number in a set is called a member or element
Sol 3a):
Advantages of measuring of central tendency:
In central tendency the large group of data is grouped into a single value
for effective business decision making.
Three common measures of central tendency are the mean, median and the mode.
They are
Sol 1 a):
The red portion of this picture shows the relation of AB
Fig: AB
The red portion of the picture bellow indicate the relation of AB
Fig: AB
The blue portion of the picture bellow represents the relation of A
Fig: A
The red portion of the picture
De Morgan's Law
DeMorgan a logician and mathematician who was acquainted with Boole, proposed two
theorem.
that are important part of boolean algebra.
They are stated as follows in eqn
form:
1) (A.B)'= A' + B'
2) (A+ B)' = A'.B'
First theorem states that
www.ncerthelp.com
KENDRIYA VIDYALAYA SANGATHAN
HYDERABAD REGION
MODEL FORMATIVEASSESSMENT- I
CLASS: VII
MARKS: 40
SUBJECT: MATHEMATICS
DURATION: 1 Hrs
General Instructions:
1. All questions are compulsory.
2. The question paper consists of 17 questions di
Training agenda
Location:
Pitman Training Institute
Date:
3rd August 2016
Time:
11:00 AM
Agenda items
1.
Create various users according to the Institutes
requirements.
2.
Training Venue Creation (Classrooms).
3.
Create trainers.
4.
Course Creation.
5.
Sch
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
Please enter your name and WGU ID number in the boxes below.
Note: In order to get credit for your work it is critical that you enter your student number correctly.
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)
,
Solution:
Let f be an isomorphic of G.
Let a1, a2 G be arbitrary.
As given
f (a1a2) = (a1a2)-1
=> f(a1)f (a2) = a2-1. a1-1 ; since f is a Homorphism,
=> a1-1 a2-1 = a2-1 a1-1, as given
=> (a1-1 a2-1)-1 = (a2-1 a1-1)-1
=> (a2-1)-1 (a1-1)-1 = (a1-1)-1 (a2-1
Solution 2a):
Contradiction: A contradictions is a formula which is "always false";
(i): p q p q
P
Q
~p
~q
p q p q
~p
p
^
o
~
r
q
q
T
T
F
F
F
T
F
T
F
F
T
F
T
F
F
T
T
F
F
T
F
F
F
T
T
T
F
F
Since Last Column contains only Fs. So it is contradiction.
(ii):
Due: 2013/05/07 before class
Homework 9
If you prefer submitting your homework via email, please
send it to TA: [email protected]
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
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: [email protected]
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: [email protected]
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: [email protected]
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: [email protected]
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: [email protected]
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
Due: 2013/04/08 before class
Homework 6
If you prefer submitting your homework via email, please
send it to TA: [email protected]
Problem 1. What is the Russells Paradox? Explain its impact on the naive
set theory.
Problem 2. Show that there is no set
Due: 2013/04/15 before class
Homework 7
If you prefer submitting your homework via email, please
send it to TA: [email protected]
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: 2013/04/23 before class
Homework 8
If you prefer submitting your homework via email, please
send it to TA: [email protected]
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: [email protected]
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