PPT MATHS (Relation and Functions) class 12.pdf

R&F with questions and answers

2. DEFINATIONDEFINATION
A relation is a connection
between elements from two
different sets. A function is a
special type of relation
where each element in the
first set is uniquely paired
with exactly one element in
the second set. Functions
have important properties
that distinguish them from
general relations.
A relation is a connection
between elements from two
different sets. A function is a
special type of relation
where each element in the
first set is uniquely paired
with exactly one element in
the second set. Functions
have important properties
that distinguish them from
general relations.

5. Some types of RelationsSome types of Relations
Empty Relation
Universal Relation
Reflexive , Symmetric ,
Transitive
Equivalence Relation
Empty Relation
Universal Relation
Reflexive , Symmetric ,
Transitive
Equivalence Relation

6. An empty relation (or void
relation) is one in which there
is no relation between any
elements of a set.
A universal (or full relation)
is a type of relation in which
every element of a set is
related to each other.

7. 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
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
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
A relation R defined on the set A is
said to be symmetric, if (x, y) is an
element of R, then (y, x) is also an
element of R. In other words, if x is
related to y, then y is also related to x.

8. Equivalence Relation If a
relation is reflexive, symmetric
and transitive at the same
time, it is known as an
equivalence relation.
On the set X = { a,b,c }, the
relation X = { ( a,a) , (b,b) , (c,c) ,
(c,b) , (b,c) } is an equivalence
relation. The following sets are
equivalence classes of this
relation: [ a ] = { a } , [ b ] = [ c ] = {
b,c } . The set of all equivalence
classes for R is { { a } , { b,c } } . This
set is a partition of the set X with
respect to R.
Example

9. Types of functions
One-One Function(injective)
Onto Function (surjective)

10. A function was defined as a relation in which each input has
exactly one output. That is, each input is mapped to a single
output value. A one-to-one function is a function in which
each input value is mapped to one unique output value. In
another way, no two input elements have the same output
value.

12. A function f : A → B is called an onto
function if each element of B is
mapped to at least one element of A.
For every b ∈ B, there is a ∈ A such that
f(a) = b.

14. A composite function is denoted by (g o f) (x) =
g (f(x)). The notation g o f is read as “g of f”.
composite functions Consider the functions f:
A→B and g: B→C. f = {1, 2, 3, 4, 5}→ {1, 4, 9, 16, 25}
and g = {1, 4, 9, 16, 25} → {2, 8, 18, 32, 50}. A = {1, 2,
3, 4, 5}, B = {16, 4, 25, 1, 9}, C = {32, 18, 8, 50,
2}.Here, g o f = {(1, 2), (2, 8), (3, 18), (4, 32), (5, 50)}.
GOF

15. Embrace the boundless potential of
mathematics, where every discovery leads to
new realms of exploration. As we continue to
push the boundaries of our understanding, we
uncover the profound and awe-inspiring
connections that underpin the very fabric of our
universe.
Embrace the boundless potential of
mathematics, where every discovery leads to
new realms of exploration. As we continue to
push the boundaries of our understanding, we
uncover the profound and awe-inspiring
connections that underpin the very fabric of our
universe.
The Endless Possibilities of
Mathematics
The Endless Possibilities of
Mathematics

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