  # Class 12 Maths Revision Notes for Relations and Functions of Chapter 1

## Class 12 Maths Revision Notes for Relations and Functions of Chapter 1 – Free PDF Download

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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 =  .
• Universal Relation: It is the relation R in X given by R = .
• 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

• function : X Y is one-one (or injective), if     .
• function : X Y is onto (or surjective), if  such that •  A function : X Y is one-one-onto (or bijective), if  is both one-one and onto.
• The composition of function : A B and : B C is the function given by  • function : X Y is invertible, if such that and • function : X Y is invertible, if and only if is one-one and onto.
• Given a finite set X, a function : X X is one-one (respectively onto) if and only if 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 is the identity element for binary operation * : , if  • An element is invertibel for binary operation * : if there exists such that where is the binary identity for the binary operation *. The element is called the inverse of and is denoted by .
• An operation * on X is commutative, if  in X.
• An operation * on X is associative, if  in X.