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# Relations and Functions class 12 Notes Mathematics

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## Relations and Functions Class 12 Notes Mathematics

Download CBSE class 12th revision notes for chapter 1 Relations and Functions in PDF format for free. Download revision notes for Relations and Functions class 12 Notes and score high in exams. These are the Relations and Functions class 12 Notes prepared by team of expert teachers. The revision notes help you revise the whole chapter 1 in minutes. Revision notes in exam days is one of the best tips recommended by teachers during exam days.

## 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 $(a*b)*c=a*(b*c),$ $\forall a,b,c$ in X.

## CBSE Class 12 Revision Notes and Key Points

Relations and Functions class 12 Notes Mathematics. CBSE quick revision note for class-12 Chemistry Physics Math’s, Biology and other subject are very helpful to revise the whole syllabus during exam days. The revision notes covers all important formulas and concepts given in the chapter. Even if you wish to have an overview of a chapter, quick revision notes are here to do if for you. These notes will certainly save your time during stressful exam days.

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