## 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**

- A
**function**: X Y is**one-one**(or**injective**), if . - A
**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- A
**function**: X Y is**invertible**, if such that and - A
**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.