Lecture 17: Continuous Functions
1
Continuous Functions
Let (X, TX ) and (Y, TY ) be topological spaces.
Definition 1.1 (Continuous Function). A function f : X Y is said to be
continuous if the inverse image of every open subset of Y is open in X. In othe
Lecture 13: Basis for a Topology
1
Basis for a Topology
Lemma 1.1. Let (X, T) be a topological space. Suppose that C is a collection of
open sets of X such that for each open set U of X and each x in U , there is an
element C C such that x C U . Then C is
Lecture 15: The subspace topology, Closed sets
1
The Subspace Topology
Definition 1.1. Let (X, T) be a topological space with topology T. If Y is a
subset of X, the collection
TY = cfw_Y U |U T
is a topology on Y , called the subspace topology. With this
Lecture 11 : Cardinality of Sets
1
Cardinality
We are interested in knowing sizes of sets. Finite sets are usually well behaved.
The difficulty starts in trying to understand infinite sets. Infinite sets have been
notoriously difficult to understand. In f
Lecture 23: Measures
1
Measures
Definition 1.1. Let (X, F) be a measurable space. A set mapping : F [0, ]
is called a measure if
i. () = 0,
ii. Countable additivity. (nN An ) =
n N of pairwise disjoint sets in F.
P
nN
(An ) for all sequences cfw_An :
Defi
Lecture 19: Connectedness
Now that we have properly defined open, closed sets and limit points in a
topological space, we can proceed to define the properties of connectedness and
compactness for arbitrary topological space. The properties of connectednes
Lecture 14: The Order Topology
1
The Order Topology
If X is a simply ordered set, there is a standard topology for X, defined using the
order relation. It is called Order Topology.
Definition: Suppose X is a set having order relation <. Given a<b X, there
Lecture 26: Dominated Convergence Theorem
Continuation of Fatous Lemma.
Corollary 0.1. If f L+ and cfw_fn L+ : n N is any sequence of functions
such that fn f almost everywhere, then
Z
Z
f 6 lim inf
fn .
X
X
Proof. Let fn f everywhere in X. That is, lim i
Lecture 2: Strategies for Proofs
1
Introduction
Thales of Miletus of sixth century BC is credited with introducing the concepts of
logical proof for abstract propositions. Euclid popularized the axiomatic system
of proofs in his book The elements, written
Lecture 6: Functions : Injectivity, Surjectivity, and
Bijectivity
1
Injectivity, Surjectivity, Bijectivity
We are interested in finding out the conditions for a function to have a left inverse,
or right inverse, or both.
Definition 1.1. Let f : A B be a f
Lecture 8: Equivalence Relations
1
Equivalence Relations
Next interesting relation we will study is equivalence relation.
Definition 1.1 (Equivalence Relation). Let A be a set and let be a relation
on A. The relation is an equivalence relation if it is re
Paper Recycling
A paper recycling plant processes box board, tissue paper, newsprint,
and book paper into pulp that can be used to produce three grades of
recycled paper. The prices per ton and the pulp contents of the four
inputs are shown in the Table b
A
B
C
1 Paper recycling
2
Cost per ton % of pulp lef
3
De-inking
$20
90%
4
Asphalt dispersion
$15
80%
5
6
Box board
Tissue paper
7
Cost per ton
$5
$6
8
Pulp content
15%
20%
9
10
11 Tons of each paper type processed into grades using de-inking
Box board
Ti
QAM-I Assignment (2014-15)
The Cutting Stock Problem
A company manufactures electrical transformers. Transformer core contains a number of metal rods of
a specified length, which varies for different models of transformers depending on the specifications.
Chapter
2
INVERSE TRIGONOMETRIC
FUNCTIONS
v Mathematics, in general, is fundamentally the science of
self-evident things. FELIX KLEIN v
2.1 Introduction
In Chapter 1, we have studied that the inverse of a function
f, denoted by f 1, exists if f is one-one
Lecture 7: Relations
1
Relation
Relation between two objects signify some connection between them. For example,
relation of one person being biological parent of another. If we take any two
people at random, say persons X and Y , then either X is a parent
Lecture 5: Functions : Images, Compositions,
Inverses
1
Functions
We have all seen some form of functions in high school. For example, we have
seen polynomial, exponential, logarithmic, trigonometric functions in calculus.
These functions map real numbers
Lecture 9: Principle of Mathematical Induction
1
Properties of Natural Numbers
Most fundamental property of natural numbers is ability to do proof by induction.
The set of natural numbers is denoted by the set N. The set N consists of a distinguished elem
Lecture 25 : Integration of non-negative functions
1
Integration of non-negative functions
Definition 1.1. Let (X, M, ) be the measurable space, we define L+ = cfw_f
F(X, [0, ]) : f is (, B[0,] ) measurable.
P
Definition 1.2. If is simple, L+ , with stan
Lecture 12 : Topological Spaces
1
Topological Spaces
Topology generalizes notion of distance and closeness etc.
Definition 1.1. A topology on a set X is a collection T of subsets of X having the
following properties.
1. and X are in T.
2. The union of the
Final Solutions
1. Find all solutions to the following system of linear equations using the
row-reduced echelon form.
x + 3y 5z =
4
x + 4y 8z =
7
3x 7y + 9z = 6
Solution. On the augmented matrix
1
3 5
1
4 8
3 7
9
4
7 ,
6
we execute following sequence of
Final
Duration: 3 hours
MTH 102, Semester 2, 2012-13
Maximum Points: 100
Name:
Roll No:
Directions: Clear and correct explanations of your solutions are necessary for full
credit. The use of electronic tools (like graphing calculators, mobile phones etc.)
Quiz II Solutions
1. Prove that every subeld of C contains Q.
Solution. Let F be a subeld of C. Then cfw_0, 1 F , and since
F is closed under addition, n 1 F , n N. Since F is a eld,
every element of F has an additive inverse, n F n N. Hence
we have that
Quiz IV Solutions
1. Let V be the space of all n n matrices over R. Let T : V V be a
1
map dened by T (A) = 2 (A + At ).
(a) Show that T is a linear map.
(b) Find a basis for Ker T and hence its dimension.
(c) Show that V = Ker T Im T .
1
1
Solution. (a)
Quiz V Solutions
1. Let V be the vector space all polynomials of degree 3 with real
coecients in one variable x.
(a) Consider the map T : V V dened by T (p(x) = p(x + a),
where a is a xed real number. Show that T is a linear operator.
B
(b) Let B be the b
Quiz VI Solutions
1. Let V be a vector space of dimension n over a eld K. Show that
cfw_v1 , v2 , . . . , vn V is a linearly independent set if and only if Det(v1 , v2 , . . . , vn ) =
0.
Solution. Suppose that cfw_v1 , v2 , . . . , vn V is a linearly d