SUMMARY OF CLASS 12 MATHS RELATIONS AND FUNCTIONS
Relation
Let A and B be two non-empty sets. Then,
a relation R from set A to set B is a subset of A × B, where A × B = {(a, b): a ∈ A and b ∈ B} is the Cartesian product of two
sets A and B. If (a, b) ∈ R, we say that a is related to b under
the relation R and we write as a R b.
Domain, Range and Co-domain of a
Relation
Let R be a relation from set A to set B,
such that R = {(a, b): a ∈ A and b ∈ B}. The set of all first elements of the ordered pairs in R is called
the domain of relation R. The set of all second elements of the ordered pairs
in R is called the range of relation R. The set B is called the co-domain of
relation R.
Types of Relations
·
Empty Relation: A relation R in a set A is called empty
relation, if no element of A is related to any element of A, i.e., R = φ ⊂ A × A.
· Universal Relation: A relation R in a set A is called universal relation, if
each element of A is related to every element of A, i.e., R = A × A.
·
A relation R in a set A is called
(i) reflexive, if (a, a) ∈ R, for every a
∈ A,
(ii) symmetric, if (a1, a2)
∈ R implies that (a2, a1) ∈ R, for all a1,
a2 ∈ A.
(iii) transitive, if (a1, a2)
∈ R and (a2, a3) ∈ R implies that
(a1, a3) ∈ R, for all a1, a2,
a3 ∈ A.
·
Equivalence Relation: A relation R in a set A is said to be an equivalence relation if R is
reflexive, symmetric and transitive.
Functions
A relation R from set X to set Y is
called as a function if for every element x ∈ X, there exists a unique element y ∈ Y, such that y = f(x).
Types of Functions
One-One (or Injective) Function
A function f : X → Y is said to be one-one (or
injective), if the images of distinct elements of X under f are
distinct, i.e., for every x1, x2 ∈ X, f(x1)
= f(x2) implies x1 = x2.
Many-One Function
A function f : X → Y is said to be many-one,
if two or more than two elements in X have the same image in Y.
Onto (or Surjective)
Function
A function f
: X → Y is said to be onto (or surjective),
if for every element y ∈ Y, there exists an
element x ∈ X such that f(x) = y.
Alternatively, if the range of f = Co-domain of f,
then f is said to be an onto function.
Bijective Function
A function f
: X → Y is said to be a bijective function,
if f is both one-one and onto.
Properties of Functions
1. If A and B are two non-empty finite sets
containing m and n elements, respectively, then the number of
functions from A to B is nm.
2. Let A and B be two sets
and f : A → B be a function.
a. If f is a one-one function, then
n(A) 
n(B).
b. If f is an onto function, then
n(A) 
n(B).
c. If f is a bijective function,
then n(A) 
n(B).