Sets Identity function

The function that associates each real number to it self is called the identity function and denoted by I
I : R \rightarrow R
I(x) = x : x \varepsilon R

The domain and range of the identity function are both equal to R. the graph is a straight line passing through the origin and inclined at an angle of 45^0 with x axis
F(x) = x
F(1) = 1
F(2) = 2
Modulus Functions:-
If f(x) = |x| = {x when x ≥ 0,
-x when x < 0} Is called the modulus function or absolute value function the domain of the modulus function is the set of all real number. & range is the set of all non negative real number R^+ = {x \varepsilon R : x ≥ 03}
(1) If x ≥ 0 the graph coincide. With the graph of
the identity functional (g= x)

If x < 0 it is coincide to the line y = -x

Properties of Modulus Function

1) For any real number x
\sqrt{(x^2 )} = |x|

2) a,b is a positive real number, then

x^2a^2 |x| ≤ a -a ≤ x ≤ a
x^2a^2 |x| ≥ a x ≤ -a or x ≥ a
x^2 < a^2 |x| < a -a < x < a x^2 > a^2 |x| > a x < -a or x > a

a^2a^2b^2 a≤ |x| ≤ b x \varepsilon [-b, -a] u [a,b]
a^2 < a^2 < b^2 a ≤ |x| ≤ b \varepsilon [-b, -a] u [a,b]

3) for real number x and y

|x+y| = |x| + |y| (x≥0 and y≥ 0) or (x< 0 and y< 0)|x-y| = |x| - |y| (x≥0 and |x|≥ |y|) or (x≤ 0 , y≤ 0 and |x| ≥ |y|)|x± y| ≤ |x| + |y||x± y| ≥ |x| - |y|

Greatest Integer Function (Floor Function)
f(x) = [x] for all xεR or ⌊x⌋ is called greatest integer function

for any real number x the smallest [x] to denote the greatest integer less than or equal to x
Domain of the greatest integer function is the set of R to all real number and the range is the set fo Z of all integers as it’s attains only integer value.

e.g. [2.75] = 2, [3]= 3, [0.74] = 0, [-7.45] = -8 etc

Properties :- if n is integer and x is a real number between n and n+1 then

[-n] = -[n]
[x + k] = [x] + k for any integer k
[-x] = -[x] -1

[x] + [-x] = \left\{\begin{array}-1, if x z\\0, x \varepsilon z \end{array}\right.
[x] – [-x] = \left\{\begin{array} 2[x]+1, if x z\\0, 2[x] \varepsilon z \end{array}\right.
[x] ≥ k = x ≥ k where k \varepsilon z
[x] ≤ k = x < k where k \varepsilon z
[x] > k = x ≥ k +1 where k \varepsilon z
[x] < k = x < k where k \varepsilon z
[x+y] = [x] + [y+x-[x] for all x,y \varepsilon R
[x] + [(x+ \frac{1}{n})] +  [(x+ \frac{2}{n})] + ………………..+ [(x+\frac{n-1}{n}] = [n x], n \varepsilon N

Smallest Integer Function
f(x) = ⌈x⌉ for all x εR
Is called the smallest integer function or the ceiling function.

The domain of the smallest integer function is the set of R of all real number and its range is the set z 0 < all integers e.g :- ⌈4.75⌉=5, ⌈-7.2⌉ = -7,⌈5⌉=5, ⌈.75⌉=1 etcProperties of Smallest Integer Function ⌈-n⌉= -⌈n⌉ ,n \varepsilon z

⌈-x⌉= -⌈x⌉+ 1,x \varepsilonR-z

⌈x+n⌉= ⌈x⌉+ n,xεR-z and n \varepsilon z
⌈x⌉+ ⌈-x⌉ = \left\{\begin{array}-1, if x z\\0, x \varepsilon z \end{array}\right.
⌈x⌉- ⌈-x⌉ = \left\{\begin{array} 2[x]+1, if x z\\0, 2[x] \varepsilon z \end{array}\right.


The function f(x) = {x} for all xεR
The symbol {x} denote the fractional part of or decimal part of x.
The domain of the fractional part function is the set of R of real number and range is the set [0,1]
f(x) = {x} = x – [x] : x εR
e.g. 3.45 = 0.45, [2.75] = 0.25, [-0.55] = 0.45, [3] = 0, [-7] = 0 etc


If f(x) = \left\{\begin{array}-\frac{|x|}{x}, x \neq 0  \\0, x= 0  \end{array}\right. .3 × .3 =.09
f(x) = \left\{\begin{array} 1, x > 0  \\0, x= 0 \\ -1 x < 0  \end{array}\right.
The domain of the signum function is the set of R (all real number) and the domain is the set

F (x) = a^x where a> 0 and a ≠ 1
If a > 1
If y = f(x) = a^x \left\{\begin{array} < 1 for x < 0  \\ =1 for x= 0 \\ >1  for x > 0  \end{array}\right.

We observe that – 2^x<3^x<4^x ……………….. : x>1
2^x = 3^x = 4^x …………………. : x = 0
2^x>3^x>4^x ……………….. : x<1

If 0 < a<1Y = f(x) = a^x decrease with the increase in x.
And y>0 : x ε R
a^x \left\{\begin{array} > 1 : x < 0  \\ =1 : x= 0 \\ <1  for x > 0  \end{array}\right.

if z<e<3 then graph

f(x) = e^x    \longrightarrow      f(x) = a^x   : a >1
f(x) = e^x  \longrightarrow        f(x) = a^x   : 0< a<1

Set Relations

A relation (“ is Religion of”) between SET A to SET B is a sub set of A×B
Where ‘R ≤ A×B
If SET A & SET B is not Vol D SET & If (a,b) \varepsalon R read as a is related to b by relation R’ if (a, b) R then a ×b (a is not related to b & y any relation R
Total Number Of Relation: If SET A and SET B not empty finite sets consisting of m and n elements then A×B consists of mn ordered pairs and total no. of sub sets of A×B is 2^{mn}
Among these 2^{mn} relations the void relation \phi & the universal relation A×B are trivial relations from A to B.

Domain And Range Of A Relation– relation to SET A to SET B all first components or co ordinates of the ordered pairs belonging to R is called the domain while the set of all second components or co-ordinates of the ordered pair in the R is called the range of R.

Relation On A Set– A non raid SET- A to relation is set (SET A) = A×A is called a relation on SET A
Inverse relation:- If SET A and SET B two sets and R is relation between them then inverse of R denoted by R^{-1}, is a relation from B to A
R^{-1} = {(b,a): (a,b) \varepsilon R}

Functions: a relation F from SET A to SET B i.e a subset of A×B is called function L or a mapping (or a map) from A to B if

(i) For each a \varepsilon A there exists bε B such that (a,b) f
(ii) (a,b) \varepsilon f and (a,c) \varepsilon f b=c

A non void set f of A×b is a function from A to B if each elements of A appears in some ordered pairs in f and no two ordered pairs in f have the first element.

If (a,b) \varepsilon f then b is called the image of a under f.
e.g. set A= (2,3,4)
set B = (3,4,5)
f1, f2, f3 is three sub set of A×B as below.
f= {(1,2,3), (3,4), (4,5)}
f2 = {(2,3), (2,4) (3,4) (4,5)}
f3= {(2,4), (3,4)}
here f1 is a function from A to B
f2 is not a function from A to B because 2 \varepsilon A has two images 3 and 4 in B.
f3 is not a function A to B because 4 varepsilon A has no image in B.
if a function F is expressed as the set of ordered pair, the domain of f is the set of all first component(elements ) of member s of f & the range of f is the set of second components of member of f.

Function As A Correspondence
If A and B is non empty sets, then a function f from set A to set B is a rule or method or correspondence. Which associate elements of set A to set B such that
(1) All elements of Set A are associated to elements in set B.
(2) An element of set A is associated to a unique element in set B.
If ‘ f ’is a function from a set A to set B then we write f:A \rigtharrow B or A \rightarrow B and read as “f is a function from A to B”.
If an elements a \varepsilon A is associated to an elements bε B then B is called the f image of a’ or image of a under f’ or the value of the function f at a’

A is called the pre image of B under function f & write b = f(a)

Description of a function:- if f:A \rightarrow B be a function such that the set A consists of a finite number of elements than f(x) described by listing the values it attains at different paints of its domain

Domain Co Domain And Range Of A Function:
IF f :A \rightarrow B , then the set A is known as the domain of f and the set B is known as the co domain of f. The set of all of image of elements of A is known as the range of f or image set of A under f and is denoted by f(A)

F(A)= {f(x): x \varepsilon A} = range of f or f(A) ≤ B
e.g. Set A= {-2,-1,0,1,2,} and B= {0,1,2,3,4,5,6}
Consider a rate f(x) = x^2
then f(-2) = (-2)^2 = 4
f(-1) = (-1)^2 = 1
f(0) = (0)^2 = 0
f(1) = (1)^2 = 1
f(2) = (2)^2 = 4

as above each elements of A is associated to a unique elements of B so, f:A \rightarrow B is given by f(x) = x^2 is a function

Here domain (f) = A {-2,-1,0,1,2}
Range (f) = {0,1,4}

Two functions f & g are said to be equal, if
(1) Domain of f = domain of g.
(2) Codomain of f = co-domain of g
(3) F(x) = g(x) for every x beginning to their common domain then function f = function g

Real Function : if domain and co domain are sub set of :
(A) (B)
The set R of all real number , function are called real function.
Domain of real function:- real functions are described. By providing the general for mald for finding the image of elements in its domain.

Range Of Real Function:- the range of a real function of a real variable. Is the set of all real value fallen by f(x) at paints in its domain.
Constant Function:- if K is a fixed real number
Then f(x) = k (x \varepsilon R)
The graph of a constant function f(x) = K is a straight line. Parallel of x – axis above or below of x axis according positive or negative value of K, if k = 0 the straight line is coincident of x axis

Set identity functions

Law of Algebra of sets

Law of Algebra of sets:

a) Impotent law- A \cap A=A and A \cup A = A
b) Identity Law – A \cap \phi = A and A \cup u=A
c) Commutative law- A \cap B = B \cap A and A \cup B = B \cup A
d) Associative law- (A \cap B) \cap C = A \cap (B \cap C)
e) Distributive law- A \cap (B \cup C) = (A \cap B)∩(A \cap C)
– A \cap (B \cap C)= (A \cup B) \cap (A \cup C)

Some Use Full Theorem

If A and B are any two sets then

a) A-B = A \cup B^'
b) B-A=B \cup A^'
c) A-B=A \longleftrightarrow A \cap B =\phi
d) (A-B)\cup B = A \cap B
e) (A-B) \cap B = \phi
f) A \leq B \longleftrightarrow B^{'} \leq A^'
g) (A-B) \cup (B-A) = (A \cup B) – (A \cap B)

Some Important Result on Number of Elements In SETS
If A,B and C are finite sets and u be the universal set then.

1. n(A \cup B) = n(A) + n(B) – n(A \cap B)
2. n(A \cup B) = n(A) + n(B) + n(B) \longleftrightarrow A,B are adjoint non vaid sets.
3. n(A-B) = n(A) – n(A \cap B)
i.e n(A-B) + n(A \cap B) = n(A)
4. n(AB) = number of elements which belong to exactly one of A or B
= n(A-B)(B-A)
= n(A-B) +n(B-A) ……………. [(A-B) and (B-A) are disjoint]
= n(A)- n(A \cap B) + n(B) – n(A \cap B)
= n(A) + n(B) – 2n(A \cap B)
5. n(A \cup B \cup C) =
n(A) + n(B) + n(c) – n(A∩B) –n(B \cap C)-n(A \cap C) + n(A \cap B \cap C)
6. number of elements in exactly two of the sets A,B,C
= n(A \cap B) + n(B \cap C) + n(C \cap A)- 3n(A \cap B \cap C)
7. number of elements in exactly on cot the sets A,B,C
= n(A) + n(B) + n(c) – 2n(A \cap B) –2n(B \cap C)-2n(A \cap C) + 3n(A \cap B \cap C)
8. n( A^{'} \cup B^' ) = n((A∩B)’) = n(u) – n(A \cup B)

Certesian Product Of Sets

Ordered Pair– an ordered pair consists of two objects or elements in a given fixed order
If A&B are any two sets then by an order pair of elements are (a,b) Where a \varepsalon A and b \varepsalon B
The position of a paint in two dimensional plane intercession coordinate is represented by an order pair (-1,5) Where x \varepsalon R and Y \varepsalon r

CARTESIAN PRODUCT OF SETS: if A and B are two non empty sets the set of all ordered pairs (a,b) such as a a \varepsalon A and b \varepsalon B is called Cartesian product of the set A and B and is denoted by AB

A×B = {a,b}: a \varepsalon A and b \varepsalon B}

Example if A= (x,y) and B= (2,3,4) find A×B, B×A, A×A
A×B = {(x,z), (x,3), (x,4), (y,z), (y,3), (y,4)}
B×A= {(z,x), (z,y), (3,x), (3,y), (4,x), (4,y)}
A×A= {(x,x), (x,y), (y,x), (y,y)}
As above (A×B) \cap (B×A) = \phi

Graphical Representation Of Cartesian Product Of Sets
If A = {2,3,4}
B = {3,4}
A×B = {(2,3) (2,4) (3,3) (3,4), (4,3), (4,4)}
To represent A×B graphical, draw to line perpendicular to each X & Y axis and then draw these pairs.
As above n(A×B) = n(A), n(B) = 2×3 = 6 Pair

A×B = \phi \longleftrightarrow A = \phi, B = \phi
A×A×A= {(a,b,c) : a,b,c A}
(a,b,c) is called an ordered triplet

(i) A× (BUC) = (A×B)U (A×C)
(ii) A×(B∩C) = (A×B)∩ (A×C)
(iii) A×(B-C) = (A×B)- (A×C)
(iv) A×B = B×A \longleftrightarrow A=B
(v) A≤B then A×A ≤ (A×B) (B×A)
(vi) A≤ B then A×C ≤ B×C
(vii) A≤B and C≤D than A×C ≤ B × D
(viii) For any set Four A,B,C,D
(A×B)∩ (C×D) = (A∩C) × (B∩D)
For any set A and B
(A×B)∩ (B×A) = (A∩B) × (B∩A)
(ix) For any three set A,B,C
A × (B’UC’) = (A×B) ∩(A×C)
A× (B’∩C’) = (A×B)U (A×C)
(x) If any two non empty sets have n elements in common. Then A×B and B×A have n^2 elements in common.
(xi) If A is non empty set and A×B = A×C
Then B=C

Set relationship

Venn Diagram

first of all a swiss mathematician euler gave an idea to represent a set by the points in a closed curve. Later on British mathematician john venn brought this idea to practice. Such the diagrams drawn to represent sets are called venn ealer diagrams or venn diagram.
In venn diagrams the universal set u is represended by points within a rectangle and its sub sets are represented by the point in closed curve (usually circle) with in the rectangle.


U (rectangle) universal Set
A- Set
B- Sub set of A
B \leq A

are show common element s of set A & set B
intersecting area = A \cup B
suppose u = {1,2,3,4,5,6,7,8,9,10}
Set A = {2,3}
Set B = {6,7,8}
A \cup B =\phi


1) Union of sets:
A & B be two sets the union of A & B is the set of all those elements which belong either to A or B or the both A & B.
A \cup B = {x:x \varepsilon A or x \varepsilon B} (A union B)
: x \varepsilon A \cup B \longleftrightarrow x \varepsilon A or x \varepsilon B
And x \varepsilon A \cup B \longleftrightarrow xA and xB
As above A \cup B = B (if A \subset B)
A \cup B = A (if B \subset A)

e.g if A = {1,2,3}
B = {1,3,5,7}
Then A \cup B = {1,2,3,5,7}

2) Intersection of sets: A & B be two sets the inter section of A & B is the set of all those elements that belong to both A & b and denoted by A \cap B

Thus A \cap B = {x:xεA and xεB}
Or x \varepsilon A \cap B \longleftrightarrow x \varepsilon A and x \varepsilon B
Or A \cap B ≤ A and A∩B ≤ B
e.g if Set A = {1,2,3}
Set B = {1,3,5,7}
Then A \cap B = {1,2,3,5,7}

3) Disjoint set: Two set A & B are said to be disjoint if A \cap B= \phi
If A\cap B \neq \phi then A and B are said to be intersecting or over capping sets.

4) Difference of sets:

Symmetric difference of two sets:- if A and b be two set s. then symmetric difference of set A and B is A \bigtriangleup b = (A-B) \cup (B-A)
= {x:x A \cap B}
e.g. Set A = {1,2,3,4,6}
Set B = {2,3,4,5,6,7}
Then A \bigtriangleup B = (A-B) \cap (B-A)
= {1,} \cap (5,7)
A \bigtriangleup B = {1,5,7}

5) complement of a set– If u is the universal set & A is a set such that A \< U than the complement of set A with respect to set u = A^' or A^C or U-A

Thus = A^' = { x \vaarepsalon u : x A}

e.g- if u = {1,2,3,4,5,6}
A= {3,4}
A^' = {1,2,5,6}

Thus u^' = \phi
\phi^'= u
(A^')' = A
A \cup A^' = u
A \cap A^' = \phi

Law of algebra of sets


A set is a well defined collection of objects.
The collection member is called number, object or element .
We shall denote sets by capital alphabets eg. A,b, ——- &
The elements by the small alphabets e.g- a,b,c,—————- z
Some reserve letters for these sets.
N: for the set of natural numbers {1,2,3,4,5,6,……………….}
Z: for the set of integers {-3,-2,-1,0,1,2,3…………………….}
Z^+ : for the set of all positive integers.
Q: for the set of all rational numbers.
T- the set of all irrational number
(x:x = \frac{m}{n} , m,n e z, {n\neq0}) (x:xe r and x\neq0)

Q^+ : for the set of all positive rational number.
R: for the set of all real numbers.
R^+: for the set of all positive real numbers.
C: for the set of all complete numbers.


Description of sets:- Set is often described in the following two ways.

1) ROSTER METHOD :- a set is described by listing elements,separated by commas, within fraces { } e.g the set of vowels of English alphabets described as (a,e,i,o,u).
2) SET SUICDER METHOD:- in this method a set is described by a characterizing property
P(x) of elements x in such a case the set is described .
{x:P(x) holded} or {x/P(x) holder} is read as
The set of all x such that P(x) holds
e.g:- A= (0,1,2,3,4,8,16,………………..) is written as A = {x^2 : xe}

EMPTY SET:- if it has no elements, said empty or void or nail set & denoted by \phi
in roster method \phi is denoted by {}
builder method = {xeR : x^2 = -z}

SINGLE TON SET a set consisting of a single element e.g {5} is a single ton set.
{x:xEN} and x^2 =9 } is a single ton set equal to {3}

FINITE SET: A finite set it is either empty set or its element can be listed or counted. By natural number (1,2,3,……….) the process of listing terminates at a certain natural number n(A) and is called ordinal number or order of a finite set

e.g. set A = (1,2,3,………………….99) Natural number less then 100
INFINITE SET : A set whose element cannot be listed by the natural number 1,2,3 …………… n for only natural number n is called an infinite set

set of all paints in a plane.
{ex er : 0 EQUIVALENT SETS: two limit sets A and B are equivalent if their cardinal numbers are same i.e n(A) = n(B)
set A = (1,2,3) set B = {a.b,c,}
Here n(A) = n(B)

EQUAL SETS: if every elements of set A is a number of set Band every elements of set B is a number of set A, are called equal sets

Set A = {1,2,5,6}
Set B = {5,6,2,1}

SUB SETS: if every elements of A is a an element of B, then A is called sub set of B and denoted by A\leq B

“A is sub set of B”
Thus A\leq B if \varepsilon A   \longrightarrow       a  \varepsilon b
\longrightarrow The symbole stands for “implies”
If A is not a sub set of B we write A \neq B

SUPER SET : If A is subset of B then B is a super set of A

Note:- 1) every set is a subset of itself A \lte B
2) empty set is subset of every set \phi \lte A
3) above 1 and 2 are called improper sub set
4) A subset A of a set B is called a proper sub set of B A \lte B
If AB then B is super set of A
If A=B then A≤B and B \lte A
5) the total number of subsets of a finite set containing n elements is 2n

A finite set A containing n elements each sub set of A have r elements
Then 0 \lte r \lte n
element is ^n C_2 then the total number of sub set of A
= ^n C_0 + ^n C_1 + ^n C_2 ………………………….. + ^n C_n = (1 + 1)^n = 2^n (binomial theorem )


1) CLOSED INTERVELS – on real line a & b two given real numbers such that a is called closed interval & written [a b]
[a , b] = {x \varepsilon r : a \lte x \lte b}

E.g. [-1,2] = {x \varepsilon r : -1 \lte x \lte 2 } is the set of all real numbers laying between -1 & 2 including -1 and 2 , clearly it is a infinite sub set of R

2) OPEN INTERVALS – on real line, a & b two given real number such that a < b , then the set of all real number x such that a < x < b is called open interval & is denoted by (a,b) or ]a,b[

3) SEMI OPENED OR SEMI CLOSED INTERVAL: on real line, a & b two given real number such that a < x ≤ b or a ≤ x < b is called semi open or semi closed interval & denoted by (a,b] or [a,b) or ]a,b], [a,b[Note:- the number b-a is called the length of any of the intervals. (a,b) [a,b] [a,b) (,b] The interval (0, \infty) denoted by set R+ of all non negative real number
The interval (- \infty,0) denoted by set R- of all negative real number
The interval (- \infty,\infty) denoted by set R of all real number

UNIVERSAL SET:a set that contains all sets under consideration i.e. it is a super set of each of the given set, is called universal set & is denoted by U Thus a set that contains all set in a given contex is called the universal.
e.g A= {1,2,3}, B = {2,4,5,6}, c= {1,3,5,7,}
then u = {1,2,3,4,5,6,7}

e.g. when we are using intervals on real line, the set R of real numbers is taken as the universal set.

POWER SET: the collection or family of all sub sets of a is called the power set & is denoted by p(A) = {s:s CA}

Let set A = {1,2,3}
Then the sub set are-
\phi}, {1}, {2}, {3},{1,2}, {1,3}, {2,3}, {1,2,3}
= P(A) (a set having n elements has 2n sub set s)
(2^3 = \varepsilon sub sets)

Venn Diagram tutorials

JavaScript variables

Variable is a user given name to a memory location where user can store values and can access it using variable name.
user can give any valid variable name to store data.
Variables can store different types of data like integer,character,string, boolean,real values etc.


for all kind of variable we use var keyword then name of variable.


After declaring variables we can assign value in variable


lets consider above variable to display in program


Java Script Variables

At the time of declaration variable can also be initialized.


JavaScript Introduction

JavaScript is a client side scripting language. Javascript code is downloaded from server and executed on clients browser.

We can include javascript statements in html file or we can create a javascript file and can import it on html files.

In html file we can use script tag to write javascript statements.


we can write script tag inside head or inside body section.



We can also put script tag inside body generally we include as last statement of body


Here document refers to body of browser and write() is a function used to write the content on browser body.

We can create a separate javascript file and can include that file in html
We have a javascript file names myJavaScript.js

Same folder having file first.html

Here we are including an external javascript file in html file the src attribute must contain path of javascript file.