Finite Intersection Property
A collection $$A$$ of subsets of a non-empty set $$X$$ is said to have the finite intersection property if every… Click here to read more
A collection $$A$$ of subsets of a non-empty set $$X$$ is said to have the finite intersection property if every… Click here to read more
A space $$X$$ is said to be locally compact (briefly $$L – $$ Compact) at $$x \in X$$ if and… Click here to read more
The process of arranging data into homogenous groups or classes according to some common characteristics present in the data is… Click here to read more
The process of placing classified data into tabular form is known as tabulation. A table is a symmetric arrangement of… Click here to read more
(1) First the data are classified and then they are presented in tables, and classification and tabulation in fact go… Click here to read more
A statistical table has at least four major parts and some other minor parts. (1) The Title (2) The Box… Click here to read more
A frequency distribution is a tabular arrangement of data into classes according to size or magnitude along with corresponding class… Click here to read more
The following steps are involved in the construction of a frequency distribution. (1) Find the range of the data: The… Click here to read more
Construct a frequency distribution with the suitable class interval size of marks obtained by 50 students of a class, which… Click here to read more
Discrete data is generated by counting, and each and every observation is exact. When an observation is repeated, it is… Click here to read more
The total frequency of all classes less than the upper class boundary of a given class is called the cumulative… Click here to read more
We have discussed the techniques of classification and tabulation that help us organize the collected data in a meaningful fashion…. Click here to read more
A simple bar chart is used to represent data involving only one variable classified on a spatial, quantitative or temporal… Click here to read more
In a multiple bars diagram two or more sets of inter-related data are represented (multiple bar diagram faciliates comparison between… Click here to read more
A sub-divided or component bar chart is used to represent data in which the total magnitude is divided into different… Click here to read more
A sub-divided bar chart may be drawn on a percentage basis. To draw a sub-divided bar chart on a percentage… Click here to read more
A pie chart can used to compare the relation between the whole and its components. A pie chart is a… Click here to read more
The theory of groups, an important part in present day mathematics, was established early in the nineteenth century in connection… Click here to read more
The concept of binary operations on a set is a generalization of the standard operations like addition and multiplication on… Click here to read more
1. Commutative Operation: A binary operation $$ * $$ over a set $$G$$ is said to be commutative if for… Click here to read more
Identity: A composition $$ * $$ in a set $$G$$ is said to admit of an identity if there exists… Click here to read more
A non-empty set $$G$$ together with at least one binary operation defined on it is called an algebraic structure. Thus… Click here to read more
An algebraic structure $$\left( {G, * } \right)$$ where $$G$$ is a non-empty set with a binary operation “$$ *… Click here to read more
A set $$G$$ with binary composition denoted multiplicatively is a group if (i) The composition is associative. (ii) For every… Click here to read more
If the commutative law holds in a group, then such a group is called an Abelian group or commutative group…. Click here to read more
Example 1: Show that the set of all integers …-4, -3, -2, -1, 0, 1, 2, 3, 4, … is… Click here to read more
Finite and Infinite Groups If a group contains a finite number of distinct elements, it is called a finite group…. Click here to read more
A binary operation in a finite set can completely be described by means of a table. This table is known… Click here to read more
Composition tables are useful in examining the following axioms in the manners explained below. Closure Property: If all the elements… Click here to read more
The identity element of a group is unique. The inverse of each element of a group is unique, i.e. in… Click here to read more
It is of common experience that a railway time table is fixed with the prevision of 24 hours in a… Click here to read more
Now here we are going to discuss a new type of addition, which is known as “addition modulo m” and… Click here to read more
Now here we are going to define another new type of multiplication, which is known as “multiplication modulo $$p$$.” It… Click here to read more
$${(a + b)^2} = {a^2} + 2ab + {b^2}$$ $${(a – b)^2} = {a^2} – 2ab + {b^2}$$ $${(a +… Click here to read more
1. If $$\frac{a}{b} = \frac{c}{d}$$ then $$ad = bc$$ 2. If $$\frac{a}{b} = \frac{c}{d}$$ then $$\frac{a}{c} = \frac{b}{d}$$ 3. If… Click here to read more
1. If $$p$$ is a positive integer and $$a \in \mathbb{R}$$, then $${a^p} = a \cdot a \cdot a \cdots… Click here to read more
1. $$y = {\log _a}x$$ if and only if $$x = {a^y}$$, $$x > 0$$ and $$y \in \mathbb{R}$$, $$a$$… Click here to read more
1. $${e^x} = 1 + x + \frac{{{x^2}}}{{2!}} + \frac{{{x^3}}}{{3!}} + \cdots $$ 2. $$\ln (1 + x) = x… Click here to read more
1. $$1 + 2 + 3 +\cdots + n = \frac{{n(n + 1)}}{2}$$ 2. $${1^2} + {2^2} + {3^2} +… Click here to read more
If $$\alpha $$ and $$\beta $$ are the roots of the Quadratic Equation $$a{x^2} + bx + c = 0$$,… Click here to read more
The $$nth$$ term $${a_n}$$ of the Arithmetic Progression (A.P) $$a,{\text{ }}a + d,{\text{ }}a + 2d, \ldots $$ is given… Click here to read more
Consider that $$a,b \in \mathbb{R}$$, then 1. $$\left| a \right| \geqslant 0$$ and $$\left| a \right| = 0 \Leftrightarrow a… Click here to read more
$$z = (a,b) = a + ib,{\text{ }}i = (0,1)$$ $$i = \sqrt { – 1} ,{\text{ }}{i^2} =- 1,{\text{}}{i^3}… Click here to read more
1) If $$\mathop {\lim }\limits_{x \to a} f(x) = l$$ and $$\mathop {\lim }\limits_{x \to a} g(x) = m$$, then… Click here to read more
If $$y = f(x)$$, then 1) $$\frac{{dy}}{{dx}} = f'(x) = \mathop {\lim }\limits_{h \to 0} \frac{{f(x + h) – f(x)}}{h}$$… Click here to read more
General Derivative Formulas: 1) $$\frac{d}{{dx}}(c) = 0$$ where $$c$$ is any constant. 2) $$\frac{d}{{dx}}{x^n} = n{x^{n – 1}}$$ is called… Click here to read more
1) \[{y_n} = \frac{{{d^n}}}{{d{x^n}}}{(ax + b)^m} = \frac{{m!}}{{(m – n)!}}{a^n}{(ax + b)^{m – n}}\] 2) \[{y_n} = \frac{{{d^n}}}{{d{x^n}}}\frac{1}{{(ax + b)}}… Click here to read more
1) \[\int {1dx = x + c} \] 2) \[\int {adx = ax + c} \] where $$a$$ is any… Click here to read more
1) \[\int {{{\sin }^n}xdx = – \frac{{\cos x{{\sin }^{n – 1}}x}}{n} + \frac{{n – 1}}{n}\int {{{\sin }^{n – 2}}xdx} }… Click here to read more
1) $$\int\limits_a^b {F'(x)dx = F(a) – F(b)} $$ is called the Fundamental Theorem of Integral Calculus. 2) \[\int\limits_a^b {f(x)dx} = –… Click here to read more