Introduction to Graphs

Let me give you introduction to Graphs,
You can measure angle sizes by using more than one scale. The degree scale is probably the most well-known scale, although the radian scale is equally as popular and useful. Although most applications deal only in one of these two scales, it is important to understand their differences and how to convert from one to the other.

You can define trig functions by using a unit circle, a circle with a radius of 1. As a point revolves around the circle, its distance from the x-axis is defined as the sine, and its distance from the y-axis is defined as the cosine. These definitions match the previous definitions in terms of a right triangle. The graphs of trigonometric functions are used to visually represent their behavior.
Hope the above explanation helped you.

Equivalence Relations

Let us study about Equivalence Relations in mathematics,
Equivalence relations permeate mathematics with several salient examples readily available:

1. Residue classes [a]N consist of all numbers congruent (equivalent) modulo N
2. a negative number is a set of all equivalent pairs (a, b) of integers with a < b, where two pairs (a, b) and (c, d) belong to the same set (equivalence class) iff a + d = b + c
3. a rational number is a set of all equivalent pairs (a, b) of integers, where two pairs (a, b) and (c, d) are equivalent iff ad = bc
4. an irrational number is an equivalence class of sequences r1, r2, r3, ... of rational numbers, where two sequences {ri} and {si} are equivalent iff they have limits as i→∞ and their limits coincide.

Examples are many and one may be inclined to think that the reason I do not cite more is that the level of abstraction of such examples rises so quickly as to make them incomprehensible to a casual reader. This is quite possible. Why make it difficult? However, I'd like to give one additional example. What's really interesting about it is that every one knows about it, learns it starting, say, at the age of 5, uses it virtually every minute. As a matter of fact, you came across it as you were reading the previous sentence.

What is this equivalence relation? This ubiquitous abstraction is the notion of integer as used in counting. Assume, in a foreign country whose language you do not speak (and starving for this reason), you ran into a McDonald's. You rush in but even in a foreign country not all of them are geniuses. The fellow there appears to react correctly to the word "hamburger" but you do not know how to say 5. What do you do? I bet you'll raise an open hand and turn the palm towards the fellow so that she'll see all your spread fingers clearly. And the fellow, an unlikely genius, will understand you perfectly well. Unlike a mathematical concept, this one is not strictly defined, but is used so universally we forget what counting and numbers are about.
Hope the above explanation helped you.

Perpendicular

Perpendicular:

Introduction:
Let us learn about perpendiculars in detail.Perpendicular exactly upright or vertical; pointing to the zenith; at right angles to the plane of the horizon; extending in a right line from any point toward the center of the earth.

At right side angles to a given line or surface; as, the line ad is perpendicular to the line bc.

A line at right side angles to the plane of the horizon; a vertical line or direction.

A line or plane falling at right side angles on another line or surface, or making equal angles with it on each side.

Finding the perpendiculars of a function:

In algebra, for any linear equation y=mx + b, the perpendiculars will all have a slope of (-1/m), the opposite reciprocal of the original slope. Perpendicular is helpful to memorize the slogan "to find the slope of a perpendicular line, flip the fraction and swap the sign." Remind that any whole number a is itself over one, and can be written as (a/1)

To find the perpendicular of a given line thats also passes through a particular point (x, y), solve the equation y = (-1/m)x + b, substituting in the known values of m, x, and y to solve for b.

Perpendicular symbol:

The perpendicular symbol is . For example, indicates that line AB is perpendicular to line CD.

In the Unicode character set, the perpendicular sign has the codepoint U+27C2 and is part of the Miscellaneous Mathematical Symbols-A range. It often looks the same as the "up tack" symbol (U+22A5), but is a different character.

Hope you like the above example of Perpendicular.Please leave your comments, if you have any doubts.

Proportional Parts of Triangles

Let us learn about proportional parts of triangles,
Consider Figure 1 of Δ ABC with line l parallel to AC and intersecting the other two sides at D and E.

Figure 1 Deriving the Side-Splitter Theorem.

You can eventually prove that Δ ABC∼ Δ DBE using the AA Similarity Postulate. Because the ratios of corresponding sides of similar polygons are equal, you can show that



Now use Property 4, the Denominator Subtraction Property.




But AB–DB = AD, and BC–BE = CE ( Segment Addition Postulate). With this replacement, you get the following proportion.




This leads to the following theorem.
Theorem (Side-Splitter Theorem): If a line is parallel to one side of a triangle and intersects the other two sides, it divides those sides proportionally.
Example 1: Use Figure 2 to find x.

Figure 2 Using the Side-Splitter Theorem.

Because DE ‖ AC in Δ ABC by Theorem 57, you get
Hope the above explanation was helpful.

Area of a triangle:

Iam sure we have all heard about Triangles,as it is one of the most common topics that we usually come across,let us now learn about the area of a Triangle. A triangle is one of the basic shapes of geometry. The triangle is a polygon and is a three-sided shape, which lies in one plane. The sum of all the angles in any triangle will be 180º. Simply it is defined as a polygon with three sides and three angles. Every triangle has three vertices (corner of the triangle), a base (any of the three sides), and an altitude (the perpendicular height from base to the opposite vertex).

Area of A Triangle:

Area of the triangle is equal to half of the product of base and its height.

A = ½ * base * height

Triangle
The best way to learn about the area of a Triangle is to solve an example problem related to area of a Triangle
Example Problem to Find Area of the Triangle:

Example 1:

Find area of the triangle with a base of 4cm and height of 6cm?

Example -triangle

Solution:

Area of the triangle(A) = ½ b*h

= [1/2] *4 *6

= [1/2] *24

Area =12cm2

Hope you like the above example of Area of A Triangle.Please leave your comments, if you have any doubts.

Areas of Sector and Segment of a Circle

Let us study what is meant by sector of a circle. Fig 1.1

The circular region enclosed
by two radii and the corresponding arc is called a
sector of the circle and the portion (or part) of the
circular region enclosed between a chord and the
corresponding arc is called a segment of the circle.
Thus, in Fig1.1, shaded region OAPB is a sector
of the circle with centre O. ∠ AOB is called the
angle of the sector. Note that in this figure, unshaded region OAQB is also a sector of
the circle. For obvious reasons, OAPB is called the minor sector and
OAQB is called the major sector. You can also see that angle of the major sector is
360° – ∠ AOB.
Now, look at Fig1.2. in which AB is a chord
of the circle with centre O. So, shaded region APB is
a segment of the circle. You can also note that
unshaded region AQB is another segment of the circle
formed by the chord AB. For obvious reasons, APB
is called the minor segment and AQB is called the
major segment.
Now let us study the formula for area of a sector of a circle. Fig 1.2

Perimeter and Area of a Circle

First of all let us understand what is perimeter,

The distance covered by travelling once around a circle is its perimeter,
usually called its circumference. the
circumference of a circle bears a constant ratio with its diameter. This constant ratio
is denoted by the Greek letter π (read as ‘pi’). In other words,
or, circumference = π × diameter
= π × 2r (where r is the radius of the circle)
= 2πr
The great Indian mathematician Aryabhata (A.D. 476 – 550) gave an approximate
value of π. He stated that π = which is nearly equal to 3.1416. It is also
interesting to note that using an identity of the great mathematical genius Srinivas
Ramanujan (1887–1920) of India, mathematicians have been able to calculate the
value of π correct to million places of decimals. As you know π is an irrational number and its decimal expansion is non-terminating and
non-recurring (non-repeating). However, for practical purposes, we generally take
the value of π as 22/7 or 3.14, approximately.
Hope the above explanation helped you, now let us learn about circle hyperbola.

Math Functions

Let me give you some introduction about math functions,
It is one of the most
important concepts in mathematics. We can, visualise a function as a rule, which produces
new elements out of some given elements. There are many terms such as ‘map’ or
‘mapping’ used to denote a function.
Definition : A relation f from a set A to a set B is said to be a function if every
element of set A has one and only one image in set B.
In other words, a function f is a relation from a non-empty set A to a non-empty
set B such that the domain of f is A and no two distinct ordered pairs in f have the
same first element.
If f is a function from A to B and (a, b) ∈ f, then f (a) = b, where b is called the
image of a under f and a is called the preimage of b under f.
The function f from A to B is denoted by f: A --> B


Example : Let N be the set of natural numbers and the relation R be defined on
N such that R = {(x, y) : y = 2x, x, y ∈ N}.
What is the domain, codomain and range of R? Is this relation a function?
Solution : The domain of R is the set of natural numbers N. The codomain is also N.
The range is the set of even natural numbers.
Since every natural number n has one and only one image, this relation is a
function. The domain of R is the set of natural numbers N. The codomain is also N.
The range is the set of even natural numbers.
Since every natural number n has one and only one image, this relation is a
function.
Hope the explanation was helpful, now let me give you more examples on Functions.

Roster or tabular form

Let me explain about Roster or tabular form, one of the method of representing a set.
In roster form, all the elements of a set are listed, the elements are being separated
by commas and are enclosed within braces { }. For example, the set of all even
positive integers less than 7 is described in roster form as {2, 4, 6}. Some more
examples of representing a set in roster form are given below :

(a) The set of all natural numbers which divide 42 is {1, 2, 3, 6, 7, 14, 21, 42}.
Note : In roster form, the order in which the elements are listed is immaterial.
Thus, the above set can also be represented as {1, 3, 7, 21, 2, 6, 14, 42}.

(b) The set of all vowels in the English alphabet is {a, e, i, o, u}.

(c) The set of odd natural numbers is represented by {1, 3, 5, . . .}. The dots
tell us that the list of odd numbers continue indefinitely.
Note : It may be noted that while writing the set in roster form an element is not
generally repeated, i.e., all the elements are taken as distinct. For example, the set
of letters forming the word ‘SCHOOL’ is { S, C, H, O, L} or {H, O, L, C, S}. Here,
the order of listing elements has no relevance.
Hope the above explanation helped you, now let me explain you about Set-builder form.

Sets and their Representations

Let us learn about Sets and their Representations,
In everyday life, we often speak of collections of objects of a particular kind, such as,
a pack of cards, a crowd of people, a cricket team, etc. In mathematics also, we come
across collections, for example, of natural numbers, points, prime numbers, etc. More
specially, we examine the following collections:
(i) Odd natural numbers less than 10, i.e., 1, 3, 5, 7, 9
(ii) The rivers of India
(iii) The vowels in the English alphabet, namely, a, e, i, o, u
(iv) Various kinds of triangles
(v) Prime factors of 210, namely, 2,3,5 and 7
(vi) The solution of the equation: x2 – 5x + 6 = 0, viz, 2 and 3.
We note that each of the above example is a well-defined collection of objects in the sense that we can definitely decide whether a given particular object belongs to a
given collection or not. For example, we can say that the river Nile does not belong to
the collection of rivers of India. On the other hand, the river Ganga does belong to this
colleciton.
We give below a few more examples of sets used particularly in mathematics, viz.
N : the set of all natural numbers
Z : the set of all integers
Q : the set of all rational numbers
R : the set of real numbers
Z+ : the set of positive integers
Q+ : the set of positive rational numbers, and
R+ : the set of positive real numbers.
Now let me explain you methods of representing a set.

Graphical Solution of Linear Inequalities in Two Variables

Let us learn about Graphical Solution of Linear Inequalities in Two Variables
we will discuss graph of a linear inequality in two variables.
We know that a line divides the Cartesian plane into two parts. Each part is
known as a half plane. A vertical line will divide the plane in left and right half planes
and a non-vertical line will divide the plane into lower and upper half planes

A point in the Cartesian plane will either lie on a line or will lie in either of the half
planes I or II. We shall now examine the relationship, if any, of the points in the plane
and the inequalities ax + by <> c.
Let us consider the line
ax + by = c, a ≠ 0,b ≠ 0
There are three possibilities namely:
(i) ax + by = c (ii) ax + by > c (iii) ax + by < c.
Hope the explanation helped you, now me give you examples on Graphical Solution of Linear Inequalities in Two Variables

Solution of a Quadratic Equation by Factorisation

Let us learn about Solution of a Quadratic Equation by Factorisation,
Consider the quadratic equation 2x2 – 3x + 1 = 0. If we replace x by 1 on the
LHS of this equation, we get (2 × 12) – (3 × 1) + 1 = 0 = RHS of the equation.
We say that 1 is a root of the quadratic equation 2x2 – 3x + 1 = 0. This also means that
1 is a zero of the quadratic polynomial 2x2 – 3x + 1.
In general, a real number α is called a root of the quadratic equation
ax2 + bx + c = 0, a ≠ 0 if a α2 + bα + c = 0.

We also say that x = α is a solution of
the quadratic equation, or that α satisfies the quadratic equation. Note that the
zeroes of the quadratic polynomial ax2 + bx + c and the roots of the quadratic
equation ax2 + bx + c = 0 are the same.

Example : Find the roots of the equation 2x2 – 5x + 3 = 0, by factorisation.
Solution : Let us first split the middle term – 5x as –2x –3x [because (–2x) × (–3x) =
6x2 = (2x2) × 3].
So, 2x2 – 5x + 3 = 2x2 – 2x – 3x + 3 = 2x (x – 1) –3(x – 1) = (2x – 3)(x – 1)
Now, 2x2 – 5x + 3 = 0 can be rewritten as (2x – 3)(x – 1) = 0.
So, the values of x for which 2x2 – 5x + 3 = 0 are the same for which (2x – 3)(x – 1) = 0,
i.e., either 2x – 3 = 0 or x – 1 = 0.
Now, 2x – 3 = 0 gives x = 3/2 and x – 1 = 0 gives x = 1.
So, x = 3/2 and x = 1 are the solutions of the equation.
In other words, 1 and 3/2 are the roots of the equation 2x2 – 5x + 3 = 0.
Verify that these are the roots of the given equation.
Note that we have found the roots of 2x2 – 5x + 3 = 0 by factorising
2x2 – 5x + 3 into two linear factors and equating each factor to zero.
Hope the above explanation was helpful to you, now let me give you more examples on Quadratic Equation by factorization.

QUADRATIC EQUATIONS

Let me help you know about Quadratic equations,

Quadratic equations come up when we deal with many real-life situations. For instance, suppose acharity trust decides to build a prayer hall having a carpet area of 300 square metres with its length one metre more than twice its breadth. What should be the length and breadth of the hall? Suppose the breadth of the hall is x metres. Then, its length should be (2x + 1) metres. We can depict this information pictorially as shown in Figure.
Now, area of the hall = (2x + 1). x m2 = (2x2 + x) m2
So, 2x2 + x = 300 (Given)
Therefore, 2x2 + x – 300 = 0

So, the breadth of the hall should satisfy the equation 2x2 + x – 300 = 0 which is a
quadratic equation.
Hope the above explanation helps you to know what is quadratic equation, now let me give some examples on Quadratic equations.

Composition of Functions and Invertible Function

Let us study the Composition of Functions and Invertible Function,
Consider the set A of all students, who appeared in Class X of a Board
Examination in 2006. Each student appearing in the Board Examination is assigned a
roll number by the Board which is written by the students in the answer script at the
time of examination. In order to have confidentiality, the Board arranges to deface the
roll numbers of students in the answer scripts and assigns a fake code number to each
roll number. Let B ⊂ N be the set of all roll numbers and C ⊂ N be the set of all code numbers. This gives rise to two functions f : A → B and g : B → C given by f (a) = the
roll number assigned to the student a and g(b) = the code number assigned to the roll
number b. In this process each student is assigned a roll number through the function f
and each roll number is assigned a code number through the function g. Thus, by the
combination of these two functions, each student is eventually attached a code number.
This leads to the following definition:
Definition Let f : A → B and g : B → C be two functions. Then the composition of
f and g, denoted by gof, is defined as the function gof : A → C given by
gof (x) = g(f (x)), ∀ x ∈ A.


Hope the above explanation gives you an idea about Composition of Functions and Invertible Function, let me give you more examples,

RELATIONS AND FUNCTIONS

Let me give you some introduction about relations and functions,

The concept of the term ‘relation’ in mathematics has been drawn from the meaning of relation
in English language, according to which two objects or quantities are related if there is a recognisable connection or link between the two objects or quantities. Let A be the set of students of Class XII of a school and B be the set of students of Class XI of the same school. Then some of the examples of relations from A to B are
(i) {(a, b) ∈ A × B: a is brother of b},
(ii) {(a, b) ∈ A × B: a is sister of b},
(iii) {(a, b) ∈ A × B: age of a is greater than age of b},
(iv) {(a, b) ∈ A × B: total marks obtained by a in the final examination is less than the total marks obtained by b in the final examination},
(v) {(a, b) ∈ A × B: a lives in the same locality as b}. However, abstracting from
this, we define mathematically a relation R from A to B as an arbitrary subset
of A × B.
If (a, b) ∈ R, we say that a is related to b under the relation R and we write as
a R b. In general, (a, b) ∈ R, we do not bother whether there is a recognisable connection or link between a and b. As seen in Class XI, functions are special kind of
relations.

Hope the above introduction helped you, now let me explain you the types of relations,