Relations and Functions
Relations Thus, domain of ⇒ ( a, c ) ∈ R, ∀a, b, c ∈ A
If A and B are two non-
= R { ( ) }
a : a, b ∈ R and range
Equivalence Relation
of R {b : ( a, b ) ∈ R }
empty sets, then a relation= A relation R on a set A is
R from A to B is a subset said to be an equivalence
Types of Relations
of A × B . relation, if it is
Empty or Void Relation: As
Representation of a simultaneously reflexive,
φ ⊂ A × A , for any set A, so
Relation symmetric and transitive on
φ is a relation on A, called
Roster form: In this form, A.
the empty or void relation.
we represent the relation Functions
Universal Relation: Since,
by the set of all ordered
A × A ⊆ A × A , so A × A is a Let A and B be two non-
pairs belongs to R .
relation on A , called the empty sets, then a function
Set-builder form: In this
universal relation. f from set A to set B is a
form, we represent the
Identity Relation: The rule which associates each
relation R from set A to
set B as
relation
= IA {(a, a ) : a ∈ A} element of A to a unique
element of B .
R {( a, b ) : a ∈ A, b ∈ B
= and is called the identity
relation on A. Domain, Codomain
the rule which relate the
Reflexive Relation: and Range of a
elements of A and B } .
A relation R on a set A is Function
Domain, Codomain and said to be reflexive
If f : A → B is a function
Range of a Relation relation, if every element
from A to B , then
Let R be a relation from a of A is related to itself.
(i) the set A is called the
non-empty set A to a non- Thus, ( a, a ) ∈ R, ∀a ∈ A ⇒ R domain of f ( x ) .
empty set B. Then, set of is reflexive.
(ii) the set B is called the
all first components or Symmetric Relation: A
codomain of f ( x ) .
coordinates of the ordered relation R on a set A is said
pairs belonging to R is to be symmetric relation (iii) the subset of B
called the domain of R , iff containing only the images
while the set of all second ( a, b ) ∈ R ⇒ (b, a ) ∈ R, ∀a, b ∈ A of elements of A is called
components or coordinates the range of f ( x ) .
i.e. aRb ⇒ bRa, ∀a, b ∈ A
of the ordered pairs
Transitive Relation: A Number of Functions
belonging to R is called the
relation R on a set A is said Let X and Y be two finite
range of R. Also, the set B
to be transitive relation, sets having m and n
is called the codomain of
iff ( a, b ) ∈ R and (b, c ) ∈ R elements respectively.
relation R.
Then each element of set X
,can be associated to any (i) x + n =n + x , n ∈ I d
function, then f ( x ) is
one of n elements of set Y. dx
(ii) −x =
− x , x ∈ I
So, total number of an odd function and if f ( x )
(iii) −x =
− x − 1, x ∉ I
functions from set X to set is an odd function, then
Y is n . m (iv) x ≥ n ⇒ x ≥ n, n ∈ I d
f (x ) is an even
(v) x > n ⇒ x ≥ n + 1,n ∈ I dx
Number of One-One
Functions function.
(vi) x ≤ n ⇒ x < n + 1,n ∈ I
(iv) The graph of an even
Let A and B are finite sets (vii) x < n ⇒ x < n, n ∈ I
function is symmetrical
having m and n elements
(viii) x + y ≥ x + y about Y -axis.
repectively, then the
(v) The graph of an odd
number of one-one Important Points To
function is symmetrical
functions from A to B is Be Remembered
about origin or symmetrical
Pm , n ≥ m
n
(i) Constant function is
in opposite quadrants.
0, n < m periodic with no
(vi) An even function can
(
n ( n − 1 )( n − 2 )… n − ( m − 1 ) , n ≥ m ) fundamental period.
= never be one-one, however
0, n<m
(ii) If f ( x ) is periodic with
an odd function may or may
Number of Onto (or 1 not be one-one.
period T , then and
Surjective) Functions f (x )
Properties of Inverse
Let A and B are finite sets
f (x ) are also periodic Function
having m and n elements
with same period T. (a) The inverse of a
respectively, then number
of onto (or surjective) (iii) If f ( x ) is periodic with bijection is unique.
(b) If f :A →B is a
functions from A to B is period T, then kf ( ax + b ) is
n m − n C1 (n − 1)m + n C2 (n − 2)m − n C3 (n − 3)m + …, n < m bijection and g : B → A is
n !,
= n m T
0, n>m periodic with period , the inverse of f , then
a
Number of Bijective fog = IB and gof = IA ,
where a, b, k ∈ R and
Functions where IA & IB are identity
a, k ≠ 0 .
Let A and B are finite sets functions on the sets A & B
having m and n elements Properties of Even respectively. If fof = I ,
and Odd Functions then f is inverse of itself.
respectively, them number
of bijective functions from (i) gof or fog is even, if (c) The inverse of a
A to B is both f and g are even or if bijection is also a bijection.
n !, if n = m f is odd and g is even or if (d) If f&g are two
=
0, if n > m or n < m f is even and g is odd. bijections
f : A → B, g : B → C & gof
Properties of (ii) gof or fog is odd, if
exist, then the inverse of
Greatest Integer both of f and g are odd.
Function gof also exists and
(iii) If f ( x ) is an even
( gof ) −1 = f −1og −1 .
,(e) The graph of f −1 (a) f (=
xy ) f ( x ) + f (y ) ⇒ f (x ) =
a kx or
obtained by reflecting the ⇒ f (x ) =
klnx f (x ) = 0
graph of f about the line
(b) f (=
xy ) f ( x ) ⋅ f (y ) (d) f ( x + y )= f ( x ) + f (y )
y =x.
⇒ f ( x ) = x n , n ∈ R or ⇒ f (x ) =
kx , where k is a
General
f (x ) = 0 constant.
If x , y are independent
(c) f ( x + y )= f ( x ) ⋅ f (y )
variables, then :
, Inverse Trigonometric Functions
Principal Values and Domains of Inverse Trigonometric/circular Functions
Function Domain Range
π π
(i) y = sin −1x −1 ≤ x ≤ 1 − ≤y ≤
2 2
(ii) y = cos −1x −1 ≤ x ≤ 1 0≤y ≤π
π π
(iii) y = tan −1x x ∈R − <y <
2 2
π π
(iv) y = cosec−1x x ≤ −1 or x ≥ 1 − ≤ y ≤ ;y ≠ 0
2 2
π
(v) y = sec−1x x ≤ −1 or x ≥ 1 0 ≤ y ≤ π;y ≠
2
(vi) y = cot −1x x ∈R 0<y <π
is aperiodic.
Properties of Inverse circular
Functions ( )
(v) y = cosec cosec−1x = x , x ≥ 1, y ≥ 1, y
P-1: is aperiodic.
y sin ( sin −1x=
(i) = ) x, x ∈ −1,1 , y ∈ − 1,1 , y ( )
(vi) y = sec sec−1x = x , x ≥ 1; y ≥ 1, y
is aperiodic. is aperiodic.
y cos ( cos −1x=
(ii) = ) x, x ∈ −1,1 , y ∈ − 1,1 , y P-2:
π π
is aperiodic. =(i) y sin −1 ( sinx ) , x ∈ R, y ∈ − , .
2 2
( )
(iii) y = tan tan −1x = x , x ∈ R, y ∈ R, y
Periodic with period 2π.
is aperiodic.
( )
(iv) y = cot cot −1x = x , x ∈ R, y ∈ R, y
Relations Thus, domain of ⇒ ( a, c ) ∈ R, ∀a, b, c ∈ A
If A and B are two non-
= R { ( ) }
a : a, b ∈ R and range
Equivalence Relation
of R {b : ( a, b ) ∈ R }
empty sets, then a relation= A relation R on a set A is
R from A to B is a subset said to be an equivalence
Types of Relations
of A × B . relation, if it is
Empty or Void Relation: As
Representation of a simultaneously reflexive,
φ ⊂ A × A , for any set A, so
Relation symmetric and transitive on
φ is a relation on A, called
Roster form: In this form, A.
the empty or void relation.
we represent the relation Functions
Universal Relation: Since,
by the set of all ordered
A × A ⊆ A × A , so A × A is a Let A and B be two non-
pairs belongs to R .
relation on A , called the empty sets, then a function
Set-builder form: In this
universal relation. f from set A to set B is a
form, we represent the
Identity Relation: The rule which associates each
relation R from set A to
set B as
relation
= IA {(a, a ) : a ∈ A} element of A to a unique
element of B .
R {( a, b ) : a ∈ A, b ∈ B
= and is called the identity
relation on A. Domain, Codomain
the rule which relate the
Reflexive Relation: and Range of a
elements of A and B } .
A relation R on a set A is Function
Domain, Codomain and said to be reflexive
If f : A → B is a function
Range of a Relation relation, if every element
from A to B , then
Let R be a relation from a of A is related to itself.
(i) the set A is called the
non-empty set A to a non- Thus, ( a, a ) ∈ R, ∀a ∈ A ⇒ R domain of f ( x ) .
empty set B. Then, set of is reflexive.
(ii) the set B is called the
all first components or Symmetric Relation: A
codomain of f ( x ) .
coordinates of the ordered relation R on a set A is said
pairs belonging to R is to be symmetric relation (iii) the subset of B
called the domain of R , iff containing only the images
while the set of all second ( a, b ) ∈ R ⇒ (b, a ) ∈ R, ∀a, b ∈ A of elements of A is called
components or coordinates the range of f ( x ) .
i.e. aRb ⇒ bRa, ∀a, b ∈ A
of the ordered pairs
Transitive Relation: A Number of Functions
belonging to R is called the
relation R on a set A is said Let X and Y be two finite
range of R. Also, the set B
to be transitive relation, sets having m and n
is called the codomain of
iff ( a, b ) ∈ R and (b, c ) ∈ R elements respectively.
relation R.
Then each element of set X
,can be associated to any (i) x + n =n + x , n ∈ I d
function, then f ( x ) is
one of n elements of set Y. dx
(ii) −x =
− x , x ∈ I
So, total number of an odd function and if f ( x )
(iii) −x =
− x − 1, x ∉ I
functions from set X to set is an odd function, then
Y is n . m (iv) x ≥ n ⇒ x ≥ n, n ∈ I d
f (x ) is an even
(v) x > n ⇒ x ≥ n + 1,n ∈ I dx
Number of One-One
Functions function.
(vi) x ≤ n ⇒ x < n + 1,n ∈ I
(iv) The graph of an even
Let A and B are finite sets (vii) x < n ⇒ x < n, n ∈ I
function is symmetrical
having m and n elements
(viii) x + y ≥ x + y about Y -axis.
repectively, then the
(v) The graph of an odd
number of one-one Important Points To
function is symmetrical
functions from A to B is Be Remembered
about origin or symmetrical
Pm , n ≥ m
n
(i) Constant function is
in opposite quadrants.
0, n < m periodic with no
(vi) An even function can
(
n ( n − 1 )( n − 2 )… n − ( m − 1 ) , n ≥ m ) fundamental period.
= never be one-one, however
0, n<m
(ii) If f ( x ) is periodic with
an odd function may or may
Number of Onto (or 1 not be one-one.
period T , then and
Surjective) Functions f (x )
Properties of Inverse
Let A and B are finite sets
f (x ) are also periodic Function
having m and n elements
with same period T. (a) The inverse of a
respectively, then number
of onto (or surjective) (iii) If f ( x ) is periodic with bijection is unique.
(b) If f :A →B is a
functions from A to B is period T, then kf ( ax + b ) is
n m − n C1 (n − 1)m + n C2 (n − 2)m − n C3 (n − 3)m + …, n < m bijection and g : B → A is
n !,
= n m T
0, n>m periodic with period , the inverse of f , then
a
Number of Bijective fog = IB and gof = IA ,
where a, b, k ∈ R and
Functions where IA & IB are identity
a, k ≠ 0 .
Let A and B are finite sets functions on the sets A & B
having m and n elements Properties of Even respectively. If fof = I ,
and Odd Functions then f is inverse of itself.
respectively, them number
of bijective functions from (i) gof or fog is even, if (c) The inverse of a
A to B is both f and g are even or if bijection is also a bijection.
n !, if n = m f is odd and g is even or if (d) If f&g are two
=
0, if n > m or n < m f is even and g is odd. bijections
f : A → B, g : B → C & gof
Properties of (ii) gof or fog is odd, if
exist, then the inverse of
Greatest Integer both of f and g are odd.
Function gof also exists and
(iii) If f ( x ) is an even
( gof ) −1 = f −1og −1 .
,(e) The graph of f −1 (a) f (=
xy ) f ( x ) + f (y ) ⇒ f (x ) =
a kx or
obtained by reflecting the ⇒ f (x ) =
klnx f (x ) = 0
graph of f about the line
(b) f (=
xy ) f ( x ) ⋅ f (y ) (d) f ( x + y )= f ( x ) + f (y )
y =x.
⇒ f ( x ) = x n , n ∈ R or ⇒ f (x ) =
kx , where k is a
General
f (x ) = 0 constant.
If x , y are independent
(c) f ( x + y )= f ( x ) ⋅ f (y )
variables, then :
, Inverse Trigonometric Functions
Principal Values and Domains of Inverse Trigonometric/circular Functions
Function Domain Range
π π
(i) y = sin −1x −1 ≤ x ≤ 1 − ≤y ≤
2 2
(ii) y = cos −1x −1 ≤ x ≤ 1 0≤y ≤π
π π
(iii) y = tan −1x x ∈R − <y <
2 2
π π
(iv) y = cosec−1x x ≤ −1 or x ≥ 1 − ≤ y ≤ ;y ≠ 0
2 2
π
(v) y = sec−1x x ≤ −1 or x ≥ 1 0 ≤ y ≤ π;y ≠
2
(vi) y = cot −1x x ∈R 0<y <π
is aperiodic.
Properties of Inverse circular
Functions ( )
(v) y = cosec cosec−1x = x , x ≥ 1, y ≥ 1, y
P-1: is aperiodic.
y sin ( sin −1x=
(i) = ) x, x ∈ −1,1 , y ∈ − 1,1 , y ( )
(vi) y = sec sec−1x = x , x ≥ 1; y ≥ 1, y
is aperiodic. is aperiodic.
y cos ( cos −1x=
(ii) = ) x, x ∈ −1,1 , y ∈ − 1,1 , y P-2:
π π
is aperiodic. =(i) y sin −1 ( sinx ) , x ∈ R, y ∈ − , .
2 2
( )
(iii) y = tan tan −1x = x , x ∈ R, y ∈ R, y
Periodic with period 2π.
is aperiodic.
( )
(iv) y = cot cot −1x = x , x ∈ R, y ∈ R, y
