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

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.