This document defines and provides examples of relations and functions. It explains that a relation connects elements between two or more sets, and provides examples of universal, identity, symmetric, inverse, reflexive, transitive, and equivalence relations. It then defines a function as a binary relation that associates every element in the first set to exactly one element in the second set. The document outlines the properties of one-to-one (injective), onto (surjective), and bijective (one-to-one and onto) functions, providing examples of each.

1. RELATIONS AND
FUNCTIONS
Made by:-Aaditya Gera
Class :-12th-A
Roll no.:-1

2. What is a RELATION?
A connection between the elements
of two or more sets is Relation .Let
A and B be two sets such that A =
{2, 5, 7, 8, 10, 13} and B = {1, 2, 3,
4, 5}. Then,
R = {(x, y): x = 4y – 3, x ∈ A and y
∈ B} (Set-builder form)
R = {(5, 2), (10, 3), (13, 4)} (Roster
form)

3. TYPES OF RELATION:-
1) UNIVERSAL
2) IDENTTY
3) SYMMETRIC
4) INVERSE
5) REFLEXIVE
6) TRANSITIVE
7) EQUIVALENCE

4. RELATIONS
Universal Relation
• A relation R in a set, say A is a universal relation if each
element of A is related to every element of A, i.e., R = A × A.
Also called Full relation. Suppose A is a set of all natural
numbers and B is a set of all whole numbers. The relation
between A and B is universal as every element of A is in set B.
Identity Relation
• In Identity relation, every element of set A is related to itself
only. I = {(a, a), ∈ A}. For example, If we throw two dice, we
get 36 possible outcomes, (1, 1), (1, 2), … , (6, 6). If we define
a relation as R: {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)}, it is
an identity relation.

5. RELATIONS
Symmetric Relation
• A relation R on a set A is said to be symmetric if (a, b) ∈ R
then (b, a) ∈ R, for all a & b ∈ A.
Inverse Relation
• Let R be a relation from set A to set B i.e., R ∈ A × B. The
relation R-1 is said to be an Inverse relation if R-1 from set B
to A is denoted by R-1 = {(b, a): (a, b) ∈ R}. Considering the
case of throwing of two dice if R = {(1, 2), (2, 3)}, R-1 = {(2,
1), (3, 2)}. Here, the domain of R is the range of R-1 and
vice-versa.

6. RELATIONS
Reflexive Relation
• If every element of set A maps to itself, the relation is
Reflexive Relation. For every a ∈ A, (a, a) ∈ R.
Transitive Relation
• A relation in a set A is transitive if, (a, b) ∈ R, (b, c) ∈
R, then (a, c) ∈ R, for all a, b, c ∈ A
Equivalence Relation
• A relation is said to be equivalence if and only if it is
Reflexive, Symmetric, and Transitive.

7. WHAT IS A FUNCTION?
• A function is a binary
relation over two sets
that associates every
element of the first
set, to exactly one
element of the
second set. Typical
examples are
functions from
integers to integers,
or from the real
numbers to real
numbers.

8. TYPES OF FUNCTIONS
1) ONE-ONE(INJECTIVE)FUNCTION
2) ONTO(SURJECTIVE)FUNCTIONS
3) ONE-ONE AND
ONTO(BIJECTIVE)FUNCTIONS

9. ONE-ONE(INJECTIVE)FUNCTION
• In this function every element
of the function's codomain is
the image of at most one
element of its domain.
• Let f be a function whose
domain is a set X. The
function f is said to be
injective provided that for all
a and b in X, whenever f(a) =
f(b), then a = b; that is, f(a) =
f(b) implies a =
b. Equivalently, if a ≠ b, then
f(a) ≠ f(b).

10. ONTO(SURJECTIVE)FUNCTIONS
• A surjective function is a
function whose image is
equal to its codomain ,if for
every element y in the
codomain Y of f, there is at
least one element x in the
domain X of f such that f(x)
= y.

11. ONE-ONE AND
ONTO(BIJECTIVE)FUNCTIONS
• Bijective is a function
between the elements of
two sets, where each
element of one set is paired
with exactly one element of
the other set, and each
element of the other set is
paired with exactly one
element of the first set.
There are no unpaired
elements.

12. THANK YOU