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,SANK
A
LP
JA
UH
A
RI
, FUNDAMENTAL’S OF MATHEMATICS
SETS
A set is a collection of well defined objects which are distinct from each other.
METHODS TO WRITE A SET :
(i) Roster Method or Tabular Method : In this method a set is described by listing elements,
separated by commas and enclose then by curly brackets.
(ii) Set builder form (Property Method) : In this we write down a property or rule which gives us
all the element of the set.
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TYPES OF SETS
Null set or empty set : A set having no element in it is called an empty set or a null set or void set, it is
denoted by φ or { }.
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Singleton set : A set consisting of a single element is called a singleton set.
Finite set : A set which has only finite number of elements is called a finite set.
Order of a finite set : The number of distinct elements in a finite set A is called the order of this set
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and denoted by O(A) or n(A). It is also called cardinal number of the set.
e.g. A = {a, b, c, d} ⇒ n(A) = 4
Infinite set : A set which has an infinite number of elements is called an infinite set.
Equal sets : Two sets A and B are said to be equal if every element of A is member of B, and every
element of B is a member of A. If sets A and B are equal, we write A = B and if A and B are not equal
then A ≠ B
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Equivalent sets : Two finite sets A and B are equivalent if their cardinal number is same i.e. n(A) =
n(B)
e.g. A = {1, 3, 5, 7}, B = {a, b, c, d} ⇒ n(A) = 4 and n(B) = 4
⇒ A and B are equivalent sets
Note - Equal sets are always equivalent but equivalent sets may not be equal
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SUBSET AND SUPERSET :
Let A and B be two sets. If every element of A is an element of B then A is called a subset of B and B is
called superset of A. We write it as A ⊂ B.
e.g. A = {1, 2, 3, 4} and B = {1, 2, 3, 4, 5, 6, 7} ⇒ A⊂ B
If A is not a subset of B then we write A ⊄ B
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PROPER SUBSET :
If A is a subset of B but A ≠ B then A is a proper subset of B. Set A is not proper subset of A so this is
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improper subset of A
Note : (i) The total number of subsets of a finite set containing n elements is 2 n.
(ii) Number of proper subsets of a set having n elements is 2n – 1.
POWER SET :
Let A be any set. The set of all subsets of A is called power set of A and is denoted by P(A)
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UNIVERSAL SET :
A set consisting of all possible elements which occur in the discussion is called a universal set and is
denoted by U.
e.g. if A = {1, 2, 3}, B = {2, 4, 5, 6}, C = {1, 3, 5, 7} then U = {1, 2, 3, 4, 5, 6, 7} can be taken as the
universal set.
SOME OPERATION ON SETS :
(i) Union of two sets : A ∪ B = {x : x ∈ A or x ∈ B}
e.g. A = {1, 2, 3}, B = {2, 3, 4} then A ∪ B = {1, 2, 3, 4}
(ii) Intersection of two sets : A ∩ B = {x : x ∈ A and x ∈ B}
e.g. A = {1, 2, 3}, B = {2, 3, 4} then A ∩ B = {2, 3}
(iii) Difference of two sets : A – B = {x : x ∈ A and x ∉ B}. It is also written as A ∩ B'.
Similarly B – A = B ∩ A' e.g. A = {1, 2, 3}, B = {2, 3, 4} ; A – B = {1}
(iv) Symmetric difference of sets : It is denoted by A ∆ B and A ∆ B = (A – B) ∪ (B – A)
(v) Complement of a set : A' = {x : x ∉ A but x ∈ U} = U – A
, e.g. U = {1, 2,........, 10}, A = {1, 2, 3, 4, 5} then A' = {6, 7, 8, 9, 10}
LAWS OF ALGEBRA OF SETS (PROPERTIES OF SETS):
(i) Commutative law : (A ∪ B) = B ∪ A ; A ∩ B = B ∩ A
(ii) Associative law : (A ∪ B) ∪ C = A ∪ (B ∪ C) ; (A ∩ B) ∩ C = A ∩ (B ∩ C)
(iii) Distributive law : A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) ; A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
(iv) De-morgan law : (A ∪ B)' = A' ∩ B' ; (A ∩ B)' = A' ∪ B'
(v) Identity law : A ∩ U = A ; A ∪ φ = A
(vi) Complement law : A ∪ A' = U, A ∩ A' = φ, (A')' = A
(vii) Idempotent law : A ∩ A = A, A ∪ A = A
SOME IMPORTANT RESULTS ON NUMBER OF ELEMENTS IN SETS :
If A, B, C are finite sets and U be the finite universal set then
n(A ∪ B) = n(A) + n(B) – n(A ∩ B)
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(i)
(ii) n(A – B) = n(A) – n(A ∩ B)
(iii) n(A ∪ B ∪ C) = n(A) + n(B) + n(C) – n(A ∩ B) – n(B ∩ C) – n(A ∩ C) + n(A ∩ B ∩ C)
(iv) Number of elements in exactly two of the sets A, B, C
= n(A ∩ B) + n(B ∩ C) + n(C ∩ A) – 3n(A ∩ B ∩ C)
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(v) Number of elements in exactly one of the sets A, B, C
= n(A) + n(B) + n(C) – 2n(A ∩ B) – 2n(B ∩ C) – 2n(A ∩ C) + 3n(A ∩ B ∩ C)
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Intervals :
Intervals are basically subsets of R and are commonly used in solving inequalities or in finding
domains. If there are two numbers a, b ∈ R such that a < b, we can define four types of intervals as
follows :
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Name Representation Discription
Open Interval (a, b) {x : a < x < b} i.e. end points are not included.
{x : a ≤ x ≤ b} i.e. end points are also included. This is possible only when
Close Interval [a, b] both a and b are finite.
Open - Closed Interval (a, b] {x : a < x ≤ b} i.e. a is excluded and b is included.
Close - Open Interval [a, b) {x : a ≤ x < b} i.e. a is included and b is excluded.
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Note : (i) (a, ∞) = {x : x > a} (ii) [a, ∞) = {x : x ≥ a} (iii) (– ∞, b) = {x : x < b}
(iv) (–∞, b] = {x : x ≤ b} (v) (– ∞, ∞) = {x : x ∈ R}
Graph of polynomial
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To plot a graph of polynomial, several sets of Points (x, y) are required.
dy
The key points are (i) stationary points ( where =0)
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dx
(ii) y-intercept ( where x is zero)
(iii) x-intercept ( where y is zero)
(iv) behaviour of polynomial at x tends to ± ∞
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and
Logarithm of A Number :
The logarithm of the number N to the base ' a ' is the exponent indicating the power to which the base ' a
' must be raised to obtain the number N. This number is designated as loga N. Hence:
logaN = x ⇔ ax = N , a > 0, a ≠ 1 & N > 0
Domain of Definition :
The existence and uniqueness of the number loga N can be determined with the help of set of
conditions, a > 0 & a ≠ 1 & N > 0.
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