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
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
256
MATHEMATICS
Appendix
2
he
d
MATHEMATICAL MODELLING
A.2.1 Introduction
bl
is
In class XI, we have learnt about mathematical modelling as an attempt to study some
part (or form) of some real-life problems in mathematical terms, i.e., the conversion of
a
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 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
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 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 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 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
Lecture 10: Recursion
1
Recursion
Consider a sequence cfw_1, 2, 4, 8, 16, . . .. This sequence is described in two ways.
First way is, let an denote the nth term of the sequence, then an = 2n1 , n N.
Second way is, let a1 = 1, and an+1 = 2an , n N. Such a
Lecture 18: 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 4: Set Operations
1
Set Operations
New sets can be constructed from existing sets by combining one or multiple of
them using set operations. This is somewhat analogous to the the construction of
natural and rational numbers. The two most basic set
Lecture 1: Informal Logic
1
Sentential/Propositional Logic
Definition 1.1 (Statement). A statement is anything we can say, write, or
otherwise express that can be either true or false.
Remark 1. Veracity of a statement doesnt depend on ones ability to ver
Lecture 24: Properties of Measures
1
Properties of Measures
We will assume (X, F) to be the measurable space throughout this lecture, unless
specified otherwise.
Definition 1.1. Let (X, F) be a measurable space. A measure : F [0, ] is
called
1. a probabil
Lecture 3: Sets
1
Sets
Logic is built on set of axioms that are assumed true. Euclid attempted to use
the axiomatic constructions for Geometry in his book The Elements. Hilbert
formalized Euclidean Geometry in 1899, with a complete set of axioms.
Set theo
Lecture 16: The subspace topology, Closed sets
1
Closed Sets and Limit Points
Definition 1.1. A subset A of a topological space X is said to be closed if the
set X A is open.
Theorem 1.2. Let Y be a subspace of X . Then a set A is closed in Y if and
only
Lecture 22: Introduction to Measure Theory
1
Introduction
The measure theory is a natural extension of the concept of measure in Euclidean
geometry, i.e., area and volume. To understand the concept of measurable sets
and measurability in more general term