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MATHEMATICS
IMPORTANT FORMULAE
AND CONCEPTS
for
Final Revision


CLASS – XII
2016 – 17



CHAPTER WISE CONCEPTS, FORMULAS FOR
QUICK REVISION




Prepared by

M. S. KUMARSWAMY, TGT(MATHS)
M. Sc. Gold Medallist (Elect.), B. Ed.

Kendriya Vidyalaya GaCHIBOWLI

Prepared by: M. S. KumarSwamy, TGT(Maths) Page - 1 -

,Prepared by: M. S. KumarSwamy, TGT(Maths) Page - 1 -

,CHAPTER – 1: RELATIONS AND FUNCTIONS

QUICK REVISION (Important Concepts & Formulae)

Relation
Let A and B be two sets. Then a relation R from A to B is a subset of A × B.
R is a relation from A to B R  A × B.

Total Number of Relations
Let A and B be two nonempty finite sets consisting of m and n elements respectively. Then A × B
consists of mn ordered pairs. So, total number of relations from A to B is 2nm.

Domain and range of a relation
Let R be a relation from a set A to a set B. Then the set of all first components or coordinates of the
ordered pairs belonging to R is called the domain of R, while the set of all second components or
coordinates of the ordered pairs in R is called the range of R.
Thus, Dom (R) = {a : (a, b) R} and Range (R) = {b : (a, b) R}.

Inverse relation
Let A, B be two sets and let R be a relation from a set A to a set B. Then the inverse of R, denoted by R–1,
is a relation from B to A and is defined by R–1 = {(b, a) : (a, b) R}.

Types of Relations
Void relation : Let A be a set. Then A × A and so it is a relation on A. This relation is called the
void or empty relation on A. It is the smallest relation on set A.

Universal relation : Let A be a set. Then A × A  A × A and so it is a relation on A. This relation is
called the universal relation on A. It is the largest relation on set A.

Identity relation : Let A be a set. Then the relation I A = {(a, a) : a A} on A is called the identity
relation on A.

Reflexive Relation : A relation R on a set A is said to be reflexive if every element of A is related to
itself. Thus, R reflexive  (a, a) R a A.


A relation R on a set A is not reflexive if there exists an element a  A such that (a, a) R.

Symmetric relation : A relation R on a set A is said to be a symmetric relation iff (a, b) R  (b, a)
R for all a, b A. i.e. aRb bRa for all a, b A.


A relation R on a set A is not a symmetric relation if there are atleast two elements a, b A such that
(a, b) R but (b, a) R.

Transitive relation : A relation R on A is said to be a transitive relation iff (a, b) R and (b, c) R 
(a, c) R for all a, b, c A. i.e. aRb and bRc aRc for all a, b, c A.

Antisymmetric relation : A relation R on set A is said to be an antisymmetric relation iff (a, b) R and
(b, a) R a = b for all a, b A.
Equivalence relation : A relation R on a set A is said to be an equivalence relation on A iff
It is reflexive i.e. (a, a) R for all a A.
It is symmetric i.e. (a, b) R (b, a) R for all a, b A.
Prepared by: M. S. KumarSwamy, TGT(Maths) Page - 2 -

, 
It is transitive i.e. (a, b) R and (b, c) R  (a, c) R for all a, b, c A.

Congruence modulo m
Let m be an arbitrary but fixed integer. Two integers a and b are said to be congruence modulo m if a – b
is divisible by m and we write a  b(mod m). Thus, a  b (mod m) a – b is divisible by m.

Some Results on Relations
If R and S are two equivalence relations on a set A, then R  S is also an equivalence relation on A.


The union of two equivalence relations on a set is not necessarily an equivalence relation on the set.

If R is an equivalence relation on a set A, then R–1 is also an equivalence relation on A.

Composition of relations
Let R and S be two relations from sets A to B and B to C respectively. Then we can define a relation SoR
from A to C such that (a, c) SoR b B such that (a, b) R and (b, c)  S. This relation is called
the composition of R and S.

Functions
Let A and B be two empty sets. Then a function 'f ' from set A to set B is a rule or method or
correspondence which associates elements of set A to elements of set B such that
(i) All elements of set A are associated to elements in set B.
(ii) An element of set A is associated to a unique element in set B.
 A function ‘f ’ from a set A to a set B associates each element of set A to a unique element of set B.
 If an element a A is associated to an element b B, then b is called 'the f image of a or 'image of a
under f or 'the value of the function f at a'. Also, a is called the preimage of b under the function f.
We write it as : b = f (a).

Domain, CoDomain and Range of a function
Let f : AB. Then, the set A is known as the domain of f and the set B is known as the codomain of f.
The set of all f images of elements of A is known as the range of f or image set of A under f and is
denoted by f (A). Thus, f (A) = {f (x) : x A} = Range of f. Clearly, f (A)  B.

Equal functions
Two functions f and g are said to be equal iff
(i) The domain of f = domain of g
(ii) The codomain of f = the codomain of g, and
(iii) f (x) = g(x) for every x belonging to their common domain.


If two functions f and g are equal, then we write f = g.

Types of Functions
(i) Oneone function (injection)
A function f : A B is said to be a oneone function or an injection if different elements of A have
different images in B. Thus, f : A  B is oneone a  b f (a)  f (b) for all a, b A  f (a) = f (b)
 a = b for all a, b  A.

Algorithm to check the injectivity of a function
Step I : Take two arbitrary elements x, y (say) in the domain of f.
Step II : Put f (x) = f (y)
Step III : Solve f (x) = f (y). If f (x) = f (y) gives x = y only, then f : A  B is a oneone function (or an
injection) otherwise not.


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