CBSE 12th Mathematics
Chapter 01 Relation and Function
Types of relations: reflexive, symmetric, transitive and equivalence relations. One to one and onto
functions.
Relation
Relation - A relation R from set A to a set B is defined as a subset of the cartesian product \(A \times B\). We can also write it as \(R \subseteq {(x, y) \in A × B : xRy}\).
Note: If \(n(A) = p\) and \(n(B) = q\) from set A to set B, then \(n(A \times B) = pq\) and number of relations = \(2^pq\).
Types of Relations
Empty relation - Definition 1 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., \(R = \phi \subset A × A\).
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., \(R = A \times A\).
Reflexive relation - if \((a, a) \in R\), for every \(a \in A\)
Symmetric relation - if \((a_1, a_2) \in R\) implies that \((a_2, a_1) \in R\), for all \(a_1, a_2 \in A\).
Transitive relation - if \((a_1, a_2) \in R\) and \((a_2, a_3) \in R\) implies that \((a_1, a_3) \in R\), for all \(a_1, a_2, a_3 \in A\).
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 function \(f : X \xrightarrow{} Y\) is defined to be one-one (or injective), if the images of distinct elements of X under f are distinct, i.e., for every \(x_1, x_2 \in X, f(x_1) = f(x_2)\) implies \(x_1 = x_2\). Otherwise, f is called many-one.
12. Types of Function:
1.One-One Function: A function f from set X to set Y is called one-one function if no two distinct elements of X have the same image in Y.
One-one function is also called an injective function.
One-one function is also called an injective function.
2.Onto Function: A function for which every element of set Y there is pre image in set X is known as Onto Function
The onto function is also known as the Surjective function.
3.One-One and Onto Function: The function f matches with each element of X with a discrete element of Y and every element of Y has a pre image in X.
The one-one and onto function is also known as the Bijective function.
4.Many to One Function: A many to one function is one that maps two or more elements of X to the same element of set Y.