Relations and Functions class 12 Notes Mathematics

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Relations and Functions Class 12 Notes Mathematics

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CBSE Class 12 Mathematics Chapter 1 Relations and Functions

TYPES OF RELATIONS:

Empty Relation: It is the relation R in X given by R = $\phi \subset$ ${\rm{X}} \times {\rm{X}}$ .
Universal Relation: It is the relation R in X given by R = ${\rm{X}} \times {\rm{X}}$ .
Reflexive Relation: A relation R in a set A is called reflexive if (a, a) ∈ R for every a ∈ A.
Symmetric Relation: A relation R in a set A is called symmetric if (, ) ∈ R implies that (, ) ∈ R, for all , ∈
Transitive Relation: A relation R in a set A is called transitive if (, ) ∈ R, and (, ) ∈ R together imply that all , , ∈ A.
EQUIVALENCE RELATION: A relation R in a set A is said to be an equivalence relation if R is reflexive, symmetric and transitive.
Equivalence Classes: Every arbitrary equivalence relation R in a set X divides X into mutually disjoint subsets (Ai) called partitions or subdivisions of X satisfying the following conditions:

All elements of Ai are related to each other for all i.
No element of Ai is related to any element of Aj whenever i ≠ j

· . These subsets () are called equivalence classes.

· For an equivalence relation in a set X, the equivalence class containing a ∈ X, denoted by [a], is the subset of X containing all elements b related to a.

**Function: A relation f: A B is said to be a function if every clement of A is correlated to a

Unique element in B.

*A is domain

* B is codomain

A function $f$ : X $\to$ Y is one-one (or injective), if $f\left( {{x_1}} \right) = f\left( {{x_2}} \right)$ $\Rightarrow$ ${x_1} = {x_2},$ $\forall$ ${x_1},{x_2} \in {\rm{X}}$ .
A function $f$ : X $\to$ Y is onto (or surjective), if $y \in {\rm{Y,}}$ $\exists x \in {\rm{X}}$ such that $f\left( x \right) = y.$
A function $f$ : X $\to$ Y is one-one-onto (or bijective), if is both one-one and onto.
The composition of function $f$ : A $\to$ B and $g$ : B $\to$ C is the function $gof:{\rm{A}} \to {\rm{C}}$ given by $gof\left( x \right) = g\left( {f\left( x \right)} \right),$ $\forall x \in {\rm{A}}{\rm{.}}$
A function $f$ : X $\to$ Y is invertible, if $\exists g:{\rm{Y}} \to {\rm{X}}$ such that $gof = {{\rm{I}}_x}$ and $fog = {{\rm{I}}_y}.$
A function $f$ : X $\to$ Y is invertible, if and only if $f$ is one-one and onto.
Given a finite set X, a function $f$ : X $\to$ X is one-one (respectively onto) if and only if $f$ is onto (respectively one-one). This is the characteristics property of a finite set. This is not true for infinite set.
A binary function * on A is a function * from A x A to A.
An element $e \in {\rm{X}}$ is the identity element for binary operation * : ${\rm{X}} \times {\rm{X}} \to {\rm{X}}$ , if $a*e=a=e*a$ $\forall a \in {\rm{X}}{\rm{.}}$
An element $e \in {\rm{X}}$ is invertibel for binary operation * : ${\rm{X}} \times {\rm{X}} \to {\rm{X}}$ if there exists $b \in {\rm{X}}$ such that $a*b*e*b*a,$ where $e$ is the binary identity for the binary operation *. The element $b$ is called the inverse of $a$ and is denoted by ${a^{ - 1}}$ .
An operation * on X is commutative, if $a*b=b*a,$ $\forall a,b$ in X.
An operation * on X is associative, if $Relations and Functions Class 12 Notes Mathematics$ $\forall a,b,c$ in X.

CBSE Class 12 Revision Notes and Key Points

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