IMMERSE 2010
Algebra Course
Problem Set 4 Solutions
1. Let G be a group, and suppose that H is the unique subgroup of G of order n. Show that H E G.
Proof (by Anne, Annette, Lisa, Matthew). Let G be a group, and suppose H is the unique subgroup
of G of or
IMMERSE 2010
Algebra Course
Problem Set 3
Work all of the following problems, and turn in a set of solutions as a group. Each group member should be
responsible for TEX-ing two problems. The final two problems are Challenge problems that we encourage
you
IMMERSE 2010
Algebra Course
Problem Set 1 Solutions
(1) Determine whether the set G is a group under the operation .
x+y
, on the set cfw_x, y R : 1 < x < 1.
(a) x y = xy+1
(b) (a, b) (c, d) = (ac bd, ad + bc), on the set R R.
Proof (by Jayna). (a) Claim:
IMMERSE 2010
Algebra Course
Problem Set 7 Solutions
1. Decide which of the following are subrings of Q:
(a) the set of all rational numbers with odd denominators (when written in lowest terms).
(b) the set of all rational numbers with even denominators (w
IMMERSE 2010
Algebra Course
Problem Set 6 Solutions
1. Classify each element of Z18 as a unit or a zero divisor. For those elements that are units, exhibit a
multiplicative inverse.
Solution (by Zach, Lisa, Tim, Sarah).
Zero Divisors: 2,3,4,6,8,9,10,12,14
4
Denite Integral
4.1
Denition, Necessary & sucient conditions
Let f : [a, b] R be a bounded real valued function on the closed, bounded interval [a, b].
Also let m, M be the inmum and supremum of f (x) on [a, b], respectively.
Denition 4.1.1. A partition
MAL 100: Calculus
Lecture Notes
1
1.1
Sequences of real numbers
Real number system
We are familiar with natural numbers and to some extent the rational numbers. While
nding roots of algebraic equations we see that rational numbers are not enough to repres
Several Variable Dierential Calculus
1
Motivations
The aim of studying the functions depending on several variables is to understand the
functions which has several input variables and one or more output variables. For example,
the following are Real valu
3
Innite Series
3.1
Denitions & convergence
Denition 3.1.1. Let cfw_an be a sequence of real numbers.
a) An expression of the form
a1 + a2 + . . . + an + . . .
is called an innite series.
b) The number an is called as the nth term of the series.
n
c) The
MAL 100: Calculus
Lecture Notes
2
2.1
Continuity, Dierentiability and Taylors theorem
Limits of real valued functions
Let f (x) be dened on (a, b) except possibly at x0 .
Denition 2.1.1. We say that lim f (x) = L if, for every real number > 0, there exist
Tutorial Sheet 1
Sequences and Series
1. Let A and B be bounded subsets of R. Show that
(i) inf(A + B) = inf A + inf B.
(ii) sup(A + B) = sup A + sup B.
2. Let q1 , q2 be two rational numbers. Then show that there exists an irrational between
them.
3. Let
1
1.1
Vector Calculus
Double integrals
Let f (x, y) be a real valued function dened over a domain I 2 . To start with, let us
R
assume that be the rectangle R = (a, b) (c, d). We partition the rectangle with node
points (xk , yk ), where
a = x1 < x2 < .,
4th Assignment, COL100
Semester II, 2015-2016
Approximation of Harmonic Number Hn
1
1
Recall that the nth Harmonic number is given as Hn = 1 + 1 + 3 + 1 + + n . It is known that as n
2
4
tends to innity, Hn converges to + ln n where is some constant. Comp