Relations and Functions Class 12 Notes Maths Chapter 1 - CBSE
Chapter:1
What are Relations and Functions ?
Relation
Let A and B be two non-empty sets. A relation from set A to set B is a subset of A×B. For example: If A={2,4,6} and B={4,9,16,25,36}, then R={(2,4),(4,16),(6,36)} is a relation from A to B as R is a subset of A×B.
Types Of Relation
One of the most important relation, which plays a significant role in Mathematics is an equivalence relation. To study equivalence relation, we first consider three types of relations, namely reflexive, symmetric and transitive.
(i) Reflexive Relation
A relation R in a given non-empty set A is called reflexive if (x,x)∈R, for all x∈A. Equivalently R is called reflexive in A if each element of A is related to itself.
(ii) Symmetric Relation
A relation R in a given non-empty set A is said to be symmetric (x,y)∈R ∈ (y,x)∈R, for all x,y ∈A.
(iii) Transitive Relation
A relation R on a given non-empty set A is said to be transitive if (x,y)∈R and (y,z)∈R ∈ (x,z)∈R, for all x,y,z∈A.
(iv) Equivalence Relation
A relation R on a given non-empty set A is said to be an equivalence if it is reflexive, symmetric and transitive.
Functions
Let A and B be two non-empty sets, then a relation f which associates each element of A with a unique element of B is called a function from A to B. It is also known as map or mapping.
Types Of Functions
(i) One to one or injective function
A function f : A B is said to be one-one or an injective if each element of set A has
unique and different image in set B. Otherwise, f is called many-one.
(ii) Onto function or subjective function
A function f : A B is said to be an onto function if and only if each element of B is
the image of some element of A under f i.e., for every y∈B there exists some x∈A such that y = f (x). Thus f is onto if and only if co-domain of f = Range of f.
(iii) One-one onto or Bijective function
A function f : A B is said to be one-one onto if and only if it is both one-one and onto i.e., each element of set A has unique and different image in set B and each element of B has pre-image in A.
