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Aspire Group of Colleges, Mandiala Tega
Class 11 Mathematics Definations
(Presented by M Zaman Yaseen Mughal)
Chapter # 1 (Number system)
Rational number: A number that can be written in the form of
, where p, q, ℤ, q ≠ 0 is called a rational number.
Irrational number: A real number that cannot be written in the form of
, where p q, , q ≠ 0 is called an irrational
number.
The real number: The field of all rational and irrational numbers is called the real numbers, or simply the "reals," and
denoted ℝ.
Terminating decimal: A decimal that has only a finite number of digits in its decimal part is called terminating decimal.
e.g. 202.04, 0.25, 0.5 example of terminating decimal.
Recurring decimal: A decimal in which one or more digits repeats indefinitely is called recurring decimal or periodic
decimal.
e.g. 0.33333 , 21.134134 … …
Note: Every terminating and recurring decimal is a rational number because it can be converted
into a common fraction.
Non-terminating, non-recurring decimal: Decimal which neither terminates nor it is recurring. It is not possible to
convert it into a common fraction. Thus non-terminating, non-recurring decimals represent an irrational number.
e.g. π = … 3.1415, we don’t have an exact decimal representation of this number.
Binary operations: A binary operation in a set A is a rule usually denoted by that assigns to any pair of elements of A to
another element of A.
e.g. two important binary operations are addition and multiplication in a set of real numbers.
Complex number: The number of the form of z=x+iy, where x,y
is called complex number. Here x is called
real part and y is called imaginary part of z .
e.g. 2,
,
.
Real plane or coordinate plane: The geometrical plane on which the coordinate system has been specified is called the
real plane or the coordinate plane.
Argand diagram: The figure representing one or more complex numbers on the complex plane is called argand diagram.
Modulus of a complex number: The modulus of a complex number is the distance from the origin of the point
representing the number. It is denoted by

or
󰇛󰇜
.
Chapter # 2 (Sets, Functions, and Groups)
Set: A set is generally described as a well-defined collection of distinct objects or a well-defined object collection of the
distinct object is called set. There are three ways to describe a set,
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Descriptive method: A method by which a set is described in words.
Example; N = the set of all-natural numbers.
Tabular method: A set may be described by listing its elements within brackets.
e.g. N = {1,2,3,4, …}
Set-builder method: In this form, we use a latter or symbol for an arbitrary element of a set and also stating the property
that is common to all members.
Example; {x|x is any nature number}
Order of a set: The number of elements in a set is called its order.
e.g. Α = {2,4} then order of Α is 2.
Equal set: Two sets A and B are said to be equal sets if each element of set A is an element of the set B both entries are
the same so A=B.
Example A = {2,4,6,8}, B = {2,4,6,8}
Equivalent set: Two sets are said to be equivalent if one to one correspondence can be established between them.
Example A = {2,4,6,8}, B={a,b,c,d}
Singleton set: A set having one element is called a singleton set.
Example Α = {2}
Null set: A set having no element is called a null set.
Example Α = Φ or {}
Finite set: A set having a finite number of elements.
Example Α = {2,4,6,8,…,100}
Infinite set: A set having an infinite number of elements.
Example Α = {2,4,6,8,…}
Subset: If each element of set Α is also an element set Β. Then Α is called sub set of Β written as:
Α Β and in the case of Β is called Β super set of Α.
(i) Empty set is a subset of every set.
(ii) Every set is a subset of itself.
Proper subset: if Α is a subset of Β and contains at least one element which is not in Α then Α is called a proper subset of
Β denoted by Α Β.
Improper subset: If a set of Β and Α = Β then Α is an improper subset of Β it follow that every set is an improper subset
of itself.
Power set: The set of all subsets of set Α is called the power set of Α, denoted by P(A ). Power set of an empty set is not
empty.
Example A = { 2,4}, then P(A ) = { Φ , {2} , {4} , {2,4} } P(A) =
.
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Universal set: Universal set is the set that contains all the elements and objects involved in the problem under
consideration or the set containing all objects or elements and of which all other sets are subsets.
Compliment of a set: The complement of a set A, denoted by
󰆒

relative to the universal set U is the set of all
elements of U, which do not belong to Α.
e.g. U= N then E = O
Deduction: To draw a general conclusion from well knows facts is called deduction.
Induction: To draw a general conclusion from a limited number of observations or experience is called induction.
Aristotelian logic: Deductive logic in which every statement is regarded as true or false is called Aristotelian logic.
Non-Aristotelian: Deductive logic in which every statement is regarded scope of third or fourth is called non-Aristotelian
logic.
Truth Table: A table to drives truth values of a given compound statement in terms of its component parts are called a
truth table.
Tautology: A statement that is true for all possible values of variable involved in it is called tautology.
e.g.
󰇛

󰇜
is a tautology.
Contradiction: A statement that is always false is called Contradiction or absurdity.
e.g.
Contingency: A statement that can be true or false depending upon the truth values of a variable.
e.g
󰇛
󰇜

󰇛

󰇜
is the contingency.
Function: Let Α and Β be two non-empty set sets. If
(1) F is a relation from Α to Β i.e. F is a subset of Α × Β.
(2) Domain of F = A (3) No two ordered pairs of F have the same 1st elements. Then F is called a function from Α to Β
and is written as F : Α → Β denoted by y=f(x).
Bijective function: (Range f = Β and 1-1) A function f which is both one to one and onto is called bijective function.
Injective function : (Rage f ≠ Β and 1-1) A function f which is both one to one and into is called
injective function.
Groupoid: A non-empty set that is closed under given Binary Operation ‘*’ is called Groupoid.
Binary operation: Any mapping of G G× into G, where G is a non-empty set, is called binary operation.
Semi group: A non-empty set is called semi group if
(i) it is closed under a given Binary operation.
(ii) The Binary operation is associative.
Monoid: A non-empty set is called Monoid.
(i) it is closed under a given Binary operation.
(ii) The Binary operation is associative.
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(iii) The set has identity element w.r.t. Binary operation
Group: A non-empty set G id called a group w.r.t Binary operation ‘*’.
(i) it is closed under a given Binary operation.
(ii) The Binary operation is associative.
(iii) The set has an identity element w.r.t. Binary operation.
(iv) Every element of G w.r.t Binary operation i.e a*a’=a’*a=e.
Abelian group: A group G under Binary operation ‘*’ is called an Abelian group if Binary operation
is commutative i.e. a*b = b*a. If a*b ≠ b*a then this is a non-abelian group under Binary operation.
Linear function: The function 󰇝󰇛󰇜
 󰇞 is called a linear function. Geometrical
representation of linear function is a straight line.
Quadratic function: The function󰇝󰇛󰇜
 󰇞 is called a quadratic function, because
it is defined by second-degree equation in x,y.
Unary Operation: A mathematical producer that changes one number into another. Or it is an
operation which is applied on a single number to give another single number .e.g
Chapter # 3 (Matrices and Determinates)
Matrix: An arrangement of different elements in the rows and columns, within square brackets is
called Matrix.
e.g.
A =
Order: Order of Matrix tells us about no of rows and columns order of a matrix = no. of rows × no. of column.
Example: A =
󰇣
󰇤
 
Row matrix: A matrix having single row is called Row Matrix.
e.g. A = [1 4 6]
Colum matrix: A matrix having single column is called column Matrix.
e. g
Square matrix: A matrix in which no of rows and columns are equal is called square matrix.
e.g. A =
󰇣
󰇤
Rectangular matrix: A matrix in which no of rows and columns are not equal is called square
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matrix. e.g.
󰇣
󰇤
Diagonal matrix: A square matrix having each of its elements excepts principle diagonal equal to
zero and at least one elements in its principle diagonal matrix.
e.g. A =
Scalar matrix: A square matrix having same elements in principle diagonal except 1 is called
scalar matrix.
e.g. A =
Unit matrix or identity matrix: Let

be a square matrix of order n. If

for all i ≠ j and

for all i = j ,
then the matrix Α is called a unit matrix or identity matrix of order n. It is denoted by
.
For example:
Null matrix or zero matrix: A square or rectangular matrix whose each element is zero, is called a null or zero matrix. It
is denoted by . Om×n .

󰇣
󰇤
Equal matrix: Two matrix are said to be equal if they are of same order with the same correspondence elements.
e.g. A =
󰇣
󰇤
, B =
󰇣
󰇤
.
Upper triangular matrix: If all elements below the principle diagonal of square matrix are zero then it is called upper
triangular matrix.
e.g. A =
Lower triangular matrix: If all elements above the principle diagonal of square matrix are zero then it is called lower
triangular matrix.
e.g. A =
Singular matrix: A square matrix Α is called singular if | A | = 0
Non-Singular matrix: A square matrix Α is called non-singular if | A | ≠ 0
Adjoint of a 2×2 matrix: The adjoint of a matrix A =
󰇣

󰇤
is denoted by adj Α and is defined as adj A =
󰇣


󰇤
Symmetric matrix: Let ‘A’ be the square matrix if
= Α then ‘Ais called symmetric matrix.
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Skew symmetric matrix: Let ‘A’ be the square matrix if t
= - Α then ‘A’ is called skew symmetric matrix.
Hermitian matrix :Let ‘A’ be the square matrix if
= Α then ‘Ais called Hermitian matrix .
Skew hermitian matrix :Let ‘A’ be the square matrix if
= -Α then ‘A’ is called skew Hermitian matrix .
Rank: Non zero row in a matrix is called rank of the matrix.
Chapter # 4 (Quadratic Equations)
Quadratic Equation: An equation of second degree polynomial in a certain variable is called Quadratic Equation.
e.g
, 
 ,
.
Equation of type 
  =0 where a =b =c ≠ 0 is called standard form of Quadratic Equation.
Solution of Quadratic Equation: (i) Factorization (ii) Quadratic Formula (iii) Completing Square.
Exponential Equation: Equations in which variable occur in exponents.
e.g

.
Reciprocal Equation: An equation which remains unchanged when x is replaced by
, is called a reciprocal equation.
Radical Equation: Equation involving radical expression of the variable is called radical equation.
Example
Remainder Theorem: If a polynomial f (x ) of degree n ≥1 is divided by (x-a) till no x term exits in the remainder then f (
a) is remainder .
Polynomial function: A polynomial in x is an expression of the form
+




+
,

where n is a non-negative integer and the coefficients
,


are real numbers. It can be considered as a polynomial function of x.
Factor Theorem: The polynomial ( x−a )is a factor of the polynomial f (x)if and only if f (x) = 0 .
Chapter # 5 (Partial Fractions)
Partial fraction: Partial fraction is an expression of a single rational function as a sum of two or more single rational
fraction.
Identity: It is an equation which holds good for all values of the variable.
Rational Fraction: The Quotient of two polynomials
󰇛
󰇜
󰇛
󰇜
where Q (x) ≠ 0 , with no common factor is called Rational
Fraction.
Proper Rational Fraction: A rational Fraction
󰇛
󰇜
󰇛
󰇜
is called. if the degree of polynomial P(x) is less degree of polynomial
Q (x).
e.g.

,


.
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Improper Rational Fraction: A Improper rational Fraction
󰇛
󰇜
󰇛
󰇜
is called improper Rational fraction. If the degree of
polynomial P (x) is greater than or equal to the degree of polynomial.
e.g. Q (x).


.



.
Conditional Equation: It is an equation which is true for particular values of variable.
e.g.2x=3 if
.
Chapter # 6 (Sequences and Series)
Sequence: Sequence is a function whose domain is subset of the set of natural numbers.
Real sequence: If all members of a sequence are real numbers, then it is called a real sequence.
Finite Sequence: If the domain of a sequence is a finite set, then the sequence is called finite sequence.
Infinite Sequence: If the domain of a sequence is an infinite set, then the sequence is called infinite sequence.
Series: The sum of an indicated number of terms in a sequence is called series.
e.g. 1+ 4 +9 +16+ 25
Arithmetic Sequence: A sequence {an} is an Arithmetic Sequence or Arithmetic progression if

is the same
number for all n Ν and n >1.
󰇛
󰇜
Arithmetic Mean: A number Α is said to be the Α.Μ. between the two numbers a and b. If a, A, b are in Α Ρ. If d is the
common difference of this Α.Ρ., then A−a =d and b− A=d .
Thus A-a=b-A

Geometric Progression: A sequence { an } is geometric sequence or geometric progression if

is the same non
zero number of all n Ν & n >1.
Geometric Mean: A number is said to be geometric means between two numbers a and b. If a, G, b are in G. P. Therefore


Harmonic Progression: A sequence of numbers is called harmonic progression or harmonic sequence if the reciprocal of
its terms are in arithmetic progression. The sequence 1,
,
,
are in harmonic sequence since there reciprocals 1,3,5,7 are
in A.P.
Harmonic means: A number H is said to be the harmonic means ( H.M ) between two numbers a and b , if a, H, b are in
H.P.
Chapter # 7 (Permutation, Combination, Probability)
Permutation: An ordering arrangement of n objects is called permutation.
Circular Permutation: The permutation of things which can be represents by the points on a circle.
Probability: Probability is the numerical evaluation of a chance that a particular event would occur.
Sample Space: The set S consisting of all possible outcome of a given experiment is called sample space.
Combination: When a selection of objects is the made without paying regard to the order of selection.
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