Chapter 1 Relations and Functions

Relation: A relation R from set X to a set Y is

defined as a subset of the cartesian product X × Y.
We can also write it as R ⊆ {(x, y) ∈ X × Y : xRy}.
Note: If n(A) = p and n(B) = q from set A to set B,
then n(A × B) = pq and number of relations = 2pq.
Types of Relation
Empty Relation: A relation R in a set X, is called
an empty relation, if no element of X is related to
any element of X,
i.e. R = Φ ⊂ X × X
Universal Relation: A relation R in a set X, is
called universal relation, if each element of X is
related to every element of X,
i.e. R = X × X
Reflexive Relation: A relation R defined on a set
A is said to be reflexive, if
(x, x) ∈ R, ∀ x ∈ A or
xRx, ∀ x ∈ R
Symmetric Relation: A relation R defined on a
set A is said to be symmetric, if
(x, y) ∈ R ⇒ (y, x) ∈ R, ∀ x, y ∈ A or
xRy ⇒ yRx, ∀ x, y ∈ R.
Transitive Relation: A relation R defined on a set
A is said to be transitive, if
(x, y) ∈ R and (y, z) ∈ R ⇒ (x, z) ∈ R, ∀ x, y, z ∈ A
or xRy, yRz ⇒ xRz, ∀ x, y,z ∈ R.
Equivalence Relation: A relation R defined on a
set A is said to be an equivalence relation if R is
reflexive, symmetric and transitive.
Equivalence Classes: Given an arbitrary
equivalence relation R in an arbitrary set X, R
divides X into mutually disjoint subsets A, called
partitions or sub-divisions of X satisfying
• all elements of Ai are related to each other,
for all i.
• no element of Ai is related to any element of
Aj, i ≠ j
• Ai ∪ Aj = X and Ai ∩ Aj = 0, i ≠ j. The subsets
Ai and Aj are called equivalence classes.
Function: Let X and Y be two non-empty sets. A
function or mapping f from X into Y written as f : X
→ Y is a rule by which each element x ∈ X is
associated to a unique element y ∈ Y. Then, f is
said to be a function from X to Y.
The elements of X are called the domain of f and
the elements of Y are called the codomain of f.
The image of the element of X is called the range
of X which is a subset of Y.
Note: Every function is a relation but every relation
is not a function.
Types of Functions
One-one Function or Injective Function: A
function f : X → Y is said to be a one-one function,
if the images of distinct elements of x under f are
distinct, i.e. f(x1) = f(x2 ) ⇔ x1 = x2, ∀ x1, x2 ∈ X
A function which is not one-one, is known as
many-one function.
Onto Function or Surjective Function: A
function f : X → Y is said to be onto function or a
surjective function, if every element of Y is image
of some element of set X under f, i.e. for every y ∈
y, there exists an element X in x such that f(x) = y.
In other words, a function is called an onto
function, if its range is equal to the codomain.
Bijective or One-one and Onto Function: A
function f : X → Y is said to be a bijective function
if it is both one-one and onto.
Composition of Functions: Let f : X → Y and g :
Y → Z be two functions. Then, composition of
functions f and g is a function from X to Z and is
denoted by fog and given by (fog) (x) = f[g(x)], ∀ x
∈ X.
Note
(i) In general, fog(x) ≠ gof(x).
(ii) In general, gof is one-one implies that f is one-
one and gof is onto implies that g is onto.
(iii) If f : X → Y, g : Y → Z and h : Z → S are
functions, then ho(gof) = (hog)of.
Invertible Function: A function f : X → Y is said
to be invertible, if there exists a function g : Y → X
such that gof = Ix and fog = Iy. The function g is
called inverse of function f and is denoted by f-1.
Note
(i) To prove a function invertible, one should prove
that, it is both one-one or onto, i.e. bijective.
(ii) If f : X → V and g : Y → Z are two invertible
functions, then gof is also invertible with (gof)-1 = f-
1
og-1
Domain and Range of Some Useful Functions

Binary Operation: A binary operation * on set X is

a function * : X × X → X. It is denoted by a * b.
Commutative Binary Operation: A binary
operation * on set X is said to be commutative, if a
* b = b * a, ∀ a, b ∈ X.
Associative Binary Operation: A binary
operation * on set X is said to be associative, if a *
(b * c) = (a * b) * c, ∀ a, b, c ∈ X.
Note: For a binary operation, we can neglect the
bracket in an associative property. But in the
absence of associative property, we cannot
neglect the bracket.
Identity Element: An element e ∈ X is said to be
the identity element of a binary operation * on set
X, if a * e = e * a = a, ∀ a ∈ X. Identity element is
unique.
Note: Zero is an identity for the addition operation
on R and one is an identity for the multiplication
operation on R.
Invertible Element or Inverse: Let * : X × X → X
be a binary operation and let e ∈ X be its identity
element. An element a ∈ X is said to be invertible
with respect to the operation *, if there exists an
element b ∈ X such that a * b = b * a = e, ∀ b ∈ X.
Element b is called inverse of element a and is
denoted by a-1.
Note: Inverse of an element, if it exists, is unique.
Operation Table: When the number of elements
in a set is small, then we can express a binary
operation on the set through a table, called the
operation table.