71 Ge2 Set Relation Mapping - Dr. Samir Kumar Bhandari

SET, RELATION
&
MAPPING
SET
• Definition:
A set is a well-defined collection of distinct objects.
• Example:
Let us consider the following collections of objects:
I. The natural numbers.
II. The trees in a garden.
III. The men in a town.
In each of the above collections, the objects have

been combined together according to some rules.
That is , these collections are well-defined.
So, all the above collections are examples of sets.
Generally sets are denoted by capital letters A,B,C,…………,X,Y,……………;
and the elements of a set are denoted by small letters a,b,c,………………
……x,y,………………….
• Finite and Infinite set :
A set consisting of a finite number of elements is called a finite set. In
Particular, a set which consists of only one element is called an one-
Point set or a singleton.
Eg. {1} , {0} are singleton sets. {1,2,3} ,{1},{0} are all finite sets.
A set consisting of infinite number of elements is an infinite set.
eg. The set of natural number={1,2,3,………………………..} .
• A set of sets is called a class or a collection or a family of sets.
• Index set:
let y be a function on a set into a set Y. we call an element i of the
domain an index and the set  an index set.
The value of the function y at i is yi and y is specified by { yi : i  }
• Subset:
if every element of a set A is an element of another set B then A is said
To be a subset of B. symbolically we write A  B or B  A.
• Power set: the collection of all subsets of a set A is called the power
set of A, is denoted by P ( A.)
Union of sets

 Let A and B be two sets. The set S consisting of all
elements which belong to at least one of the sets A
and B is called the sum or union or join of A and B. It
is denoted by S  A B .
A B
A B  {x : x  A  x  B}
PRODUCT OR INTERSECTION
• Let A and B be two sets. The set P consisting

of all elements common to A and B is called
the product or intersection or meet of the sets
A and B. It is denoted by P  A B .
A B  {x : x  A  x  B}
DISJOINT OF SETS
• Two sets A and B are said to be disjoint if

there is no element common to A and B. This
is expressed by A B   .
A B 
DIFFERENCE OF SETS
• Let A and B be two sets. Then the difference

of A and B, denoted by A-B, is the set of those
elements of A which do not belong to B.
Similarly, we can define B-A.
A  B  {x : x  A  x  B} B  A  {x : x  B  x  A}
COMPLEMENT OF A SET
• If A is a set then the complement of A is the

Set of all those elements of U which does not
Belong to A, i.e., U-A is defined to be the
Complement of A. It is denoted by CA.
U  A  A  A  {x : x  A  x  U }
c
SYMMETRIC DIFFERENCE
• The symmetric difference of two sets A and

B is defined to be the set ( A  B) ( B  A)  ( A B)  ( A B)
It is denoted by AB . Symmetric difference
satisfies commutative and associative law.
A B
B A
PARTITION OF A SET
• Let A be a non-empty set and P be a

collection of non-empty subsets of A.
Then P is called a partition of A, if the following
properties are satisfied:
(i) For all Ai , Aj  P ,either Ai  Aj or, Ai Aj   .
(ii) A  Ai .
Ai P
X A B C D E F
Cartesian Product
 The cartesian product of two non-empty
Sets X and Y, denoted by X  Y , is the set
{( x, y ) : x  X , y  Y } .
Example:
Let X={1,2} and Y={3,4} then X  Y  {(1,3),(1,4),(2,3),(2,4)}
Similarly Y  X  {(3,1),(3,2),(4,1),(4,2)}
The idea of the cartesian product of sets may
Be extended to any finite number of non-
Empty sets. For any sets X 1 , X 2 ,....., X n ;
n
X 1  X 2  .....  X n   X i  {( x1 , x2 ,....., xn ) : xi  X i , i  1, 2,...., n}
i 1
RELATION
A binary relation or simply a relation R

from a set A into a set B is a subset of A  B
. In other words, any subset R of the
Cartesian product A  B is a relation

from A to B.
Example in Real life

 Let A denote the set of names of all the districts in
West Bengal and B= {0,1,2,3,4,…………….} . With each
district x in A, let us associate the number of colleges,
say n, in that district, as in the year 2002. Then
R={(x,n)| x  A and n is the number of colleges in x, as
In the year 2002}  A  B and hence R defines a
Relation from A into B.
RELATION
Reflexive Symmetric Transitive
Let A be a set and R be a relation on A. If for alla, b, c  A ,When

R be a relation on If for all a, b  A , -ever aRb and bRc
A. If for all Whenever aRb holds, hold aRc must also
a  A, aRa. bRa must also hold. hold.
Let R be a relation Let L be a set of lines Let R be a relation

On N such that In a plane and R be On N such that
aRb iff. a|b. a relation on L such aRb iff. a|b.
Then R is reflexive. That two lines are Then R is reflexive.
perpendicular to
Each other.
EQUIVALENCE RELATION
A relation R on a set A is called an equivalence relation
if R is
reflexive, symmetric and transitive.
EXAMPLE
Let L be a set of lines in a plane. R be a relation on the
set L. R={(a, b): a || b i.e. ‘a’ line is parallel to ‘b’ line on
the set L} . Here R is an equivalence relation on L i.e.,
It is reflexive, symmetric and transitive relation on L.
F U N DA M E N T A L T H E O R E M O N A
R ELAT I O N
 Statement:
An equivalence relation  on a set S
determines a partition of S .
Conversely, each partition of S yields
an equivalence relation on S .
Mapping/Function
 Definition:
Let A and B be two non-empty sets. A mapping
(function) f from A to B is a rule that assigns to each
Element x of A there is exactly one element y in B.
Symbolically we can write .
Example: let A = The set of names of all those countries that

qualified to the finals of the FIFA World Cup 2002 and B =The set of
names of all the football players of those countries enrolled for this event.
Let us define a relation f by associating with each country in A, the name
Of the captain of its football team from B. Then clearly, f  A  B and
since every country in A has one(only one) captain of its football team,
Hence f is a function from A to B.
EVERY MAPPING IS A RELATION BUT EVERY
RELATION MAY NOT BE A MAPPING

 Let A={1,2,3} and B={1,4,9,5} be two sets. Here we
Define the product
A  B  {(1,1),(1, 4),(1,9),(1,5),(2,1),(2, 4),(2,9),(2,5),(3,1),(3, 4),(3,9),(3,5)}
We know that any kind subset of the Cartesian product
of two sets is a relation from A to B. Here we take two
Subsets R1  {(1,1), (2, 4), (3, 9)} and R2  {(1, 2), (2,5), (3,9)} ,
Both are the relations from A to B but the first relation
Is also a mapping and second relation is only a relation.
Injective Mapping
A mapping f:AB is said to be injective mapping
if for each pair of distinct elements of A,
their f-images are distinct.
Surjective Mapping
A mapping f:AB is said to be surjective
mapping if f(A)=B.
Bijective Mapping
A mapping f:AB is said to be bijective mapping
If f is both injective and surjective mapping.
Example of Mapping

 We consider f: Z Z by defining f(x)=4x-5 for all x
belongs to Z. By the property of injective mapping we
can prove that it is injective.
 Consider a map f: Z Z defining f(x)=|x| for all x
Belongs to Z. By the property of surjective mapping we
Can prove that it is surjective.
 We consider f: R R by defining f(x)=4x-5 for all x
belongs to Z. By the property of bijective mapping we
can prove that it is bijective.
Some Well –known Functions with Graph

 Signum function.
 Dirichlet’s function.
 Modulus function.
 Greatest Integer function.
 Monotonic function.
 Periodic function.
 Exponential function.
 Logarithmic function.
 Trigonometric function.
Signum & Dirichlet’s Function
The signum function f is defined by  1, x  0

f ( x )   0, x  0
 1, x  0

The Dirichlet’s function is defined by 1, when x is rational.

y  f ( x)  
The function f defined above has its domain ( , )
 0, when x is irrational.
. The value of f jump
infinitely often from 1 to 0 and back, in any interval of x, however small.
Modulus Function
The modulus function is defined by y  x in (, ) .
The graph of this function consists of two half-lines
y  x , y   x bisecting the axes and meeting at the origin.
Greatest Integer Function
For any real number x , [ x]denotes the greatest integer n
such that n  x . For every real x , an integer n s.t.
n  1  x  n . The function f defined y  f ( x)  [ x] is called
the greatest integer function. All these segments lie on the
third quadrant. The lines look like steps; hence it is called
as STEP FUNCTION.
Monotonic Function
Let us consider two points x1 , x2 in the domain of definition D of the
Function f and let x1  x2 . Then the function f is increasing (decreasing)
In D if f ( x1 )  f ( x2 )( f ( x1 )  f ( x2 )) . If the function is monotone
Then it is either an increasing or decreasing function and strictly
monotone then “equal to ” sign be omitted from above .
Monotonic
decreasing
Periodic Function
For any function f , a real number  is called a period if whenever
x  dom. f , then x   also belongs to dom. f and f ( x)  f ( x   ) .
The well-known periodic functions are y  sin x, y  cos x having
Periods 0, 2 , 4 ,............. .
Exponential Function
An exponential function has the form y  a x
. When a  0 , a  1
, y is defined for all x in ( , ) . If a  0 , we are to restrict the
domain (x should be of the form p/q with q odd).
Logarithmic Function
strictly increases in (, ) , hence inverse of y  a
x
Since a x
( a  1)
exists. We call it Logarithmic function x  log a y (0  y  ,   x  )
or, y  log a x (0  x  ,   y  ) . When a  e , the inverse
Function is called the natural logarithm of x, written as y  log e x
 log x  ln x(0  x  ,  x y  ) . We have shown the graph of
y  log x and its inverse e .
e
Trigonometric Function
The trigonometric (or circular )functions are
y  sin x , y  cos x , y  tan x
and their reciprocals y  cos ecx , y  sec x , y  cot x .
We shall always take the radian measure of the angle as the argument x .
This is a graph of cosx and secx.
This is a graph of tanx and cotx

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SET, RELATION

In each of the above collections, the objects have

• Let A and B be two sets. The set P consisting

• Two sets A and B are said to be disjoint if

• Let A and B be two sets. Then the difference

• If A is a set then the complement of A is the

• The symmetric difference of two sets A and

• Let A be a non-empty set and P be a

A binary relation or simply a relation R

Cartesian product A  B is a relation

Reflexive Symmetric Transitive

Let A be a set and R be a relation on A. If for alla, b, c  A ,When

Let R be a relation Let L be a set of lines Let R be a relation

Example: let A = The set of names of all those countries that

The Dirichlet’s function is defined by 1, when x is rational.

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