discrete mathematics Relation and Function presentation

1. RELATION AND
FUNCTION
NAME:- BISWAJIT GHOSH
STUDENT CODE:- BWU/BTA/21/179
SECTION:-C
SUBJECT:- DISCRETE MATHEMATICS
SUBJECT CODE:- PCC-CSM405
PROGRAM NAME:- B.TECH CSE(AI-ML)

2. Relations Definition:-
 A relation is a set of ordered pairs.
 The domain of a relation is the set of all first
coordinates (x-coordinates) from the ordered pairs.
 The range of a relation is the set of all second
coordinates (y-coordinates) from the ordered pairs.
 The graph of a relation is the set of points in the
coordinate plane corresponding to the ordered pairs in
the relation.

4. TYPES OF RELATION
 Empty Relation:-
An empty relation (or void relation) is one in which
there is no relation between any elements of a set. For
example, if set A = {1, 2, 3} then, one of the void
relations can be R = {x, y} where, |x – y| = 8. For empty
relation,
R = φ ⊂ A × A
 Universal Relation:-
A universal (or full relation) is a type of relation in
which every element of a set is related to each other.
Consider set A = {a, b, c}. Now one of the universal
relations will be R = {x, y} where, |x – y| ≥ 0. For

5.  Identity Relation:-
In an identity relation, every element of a set is related
to itself only. For example, in a set A = {a, b, c}, the
identity relation will be I = {a, a}, {b, b}, {c, c}. For
identity relation,
I = {(a, a), a ∈ A}
 Inverse Relation:-
Inverse relation is seen when a set has elements which are
inverse pairs of another set. For example if set A = {(a,
b), (c, d)}, then inverse relation will be R-1 = {(b, a),
(d, c)}. So, for an inverse relation,
R-1 = {(b, a): (a, b) ∈ R}.

6.  Reflexive Relation:-
In a reflexive relation, every element maps to itself. For example, consider a
set A = {1, 2,}. Now an example of reflexive relation will be R = {(1, 1), (2, 2), (1,
2), (2, 1)}. The reflexive relation is given by-
(a, a) ∈ R
 Symmetric Relation:-
In a symmetric relation, if a=b is true then b=a is also true. In other words, a
relation R is symmetric only if (b, a) ∈ R is true when (a,b) ∈ R. An example of
symmetric relation will be R = {(1, 2), (2, 1)} for a set A = {1, 2}. So, for a
symmetric relation,
aRb ⇒ bRa, ∀ a, b ∈ A

7.  Transitive Relation:-
For transitive relation, if (x, y) ∈ R, (y, z) ∈ R, then
(x, z) ∈ R. For a transitive relation,
aRb and bRc ⇒ aRc ∀ a, b, c ∈ A
 Equivalence Relation:-
If a relation is reflexive, symmetric and transitive at
the same time, it is known as an equivalence relation.

8. FUNCTION
A function is a relation defined from set A to set B such that each and every
element of set A is uniquely related to an element of set B.
Pre-image and Image of a function:-
A function f: A ⇢ B such that for each a ∈ A, there exists a unique b ∈ B such
that (a, b) ∈ R then, a is called the pre-image of f and b is called the image of
f.

9. TYPES OF FUNCTION
One-One function ( or Injective Function):-
 A function in which one element of the
domain is connected to one element of the
codomain.
 A function f: A ⇢ B is said to be a one-one
(injective) function if different elements of A
have different images in B.
Many-One function:-
 A function f: A ⇢ B is said to be a many-one
function if two or more elements of set A have
the same image in B.
 A function f: A ⇢ B is a many-one function if it
is not a one-one function.

10. Onto function( or Surjective Function):-
 A function f: A -> B is said to be onto
(surjective) function if every element of B is
an image of some element of A i.e. f(A) = B
or range of f is the codomain of f.
 A function in which every element of the
codomain has one pre-image.
Into Function:-
 A function f: A ⇢ B is said to be an into a
function if there exists an element in B with
no pre-image in A.
 A function f: A⇢ B is into function when it
is not onto.

11. One-One Correspondent function(
or Bijective Function or One-One
Onto Function):-
A function which is both one-one
and onto (both injective and
surjective) is called one-one
correspondent(bijective)
function.
One-One Into function:-
A function that is both one-one
and into is called one-one into
function.

12.  Many-one onto function:
A function that is both many-one
and onto is called many-one onto
function.
 Many-one into a function:
A function that is both many-one
and into is called many-one into
function.

13. THANK YOU