Set Theory, Chapter1, Class11

Basic concepts and formulas based on set theory chapter 1 cbse class 11. Useful and important points based on set theory. Revision notes of chapter 1 class 11 set theory.
Introduction Of Set Theory

The concept of sets
serves as a fundamental part of the present day mathematics. Sets are used to
define the concepts of relations and functions. The study of geometry
sequences, probability etc. requires the knowledge of sets. The theory of sets
developed by German Mathematician Georg Cantor(1845  1918)
Few examples of sets
are given below
N : the set of all
natural numbers.
W : the set of all
whole numbers.
Z : the set of all
integers.
Q : the set of all
rational numbers.
R : the set of all real
numbers.
Z^{+} : The set
of positive integers.
Q^{+} : The set of positive
rational numbers.
R^{+} : The set of positive real
numbers.
Definition of set
A collection of well defined and different objects is
called set.
Sets are denoted by capital letters like A, B, C,……..etc.
Elements / objects / members of a set
are represented by small letters like a, b, c, …….etc.
If a is the element of set A , then we say
that a ∊ A and is read as a belongs to set A. If a is not the element
of A then we write a ∉ A and is read as a does not belongs to A.
The symbol "∊"
is called epsilon and it
means belongs to
Method of representation of sets.

Method of representation of sets.
There are two methods of representing a
set
a) Roster form or tabular form or Enumeration form :
In roster form all the elements are
listed. The elements are being separated by commas and are enclosed within the
braces { }. For example the set of all
vowels in English alphabet are written as
A = { a, e, i, o, u}.
While writing the elements in roster form
the elements are not be repeated i.e. elements are taken as distinct.
Order of the elements of the set may be
different. The above set may be written as A = { o, e, a, i, u}
Note :Set of all real numbers is a set but cannot be written in the roster form.
b) Setbuilder form or selector form or Rule method :
In set builder form all the elements of a
set possess a single common property, which is not possessed by any element out
side the set.
For example the set of all vowels in English
alphabet are written as
A = {x : x is a vowel in English alphabet} and it is read as
Set A is the set of all x such that x is the vowel of English alphabet.
Symbol " : " is named as colon and is read as such that.
Empty set or Null set or Void set :
A set which does not contain
any element is called empty set or void set or null set. Empty set is denoted by ɸ
Finite set
 A set which is empty or consists of finite
number of elements is called finite set. or
 If the number of elements of a set are countable then it is called finite set. or
 A set is said to be finite if it has finite number of elements.
For Example: A = {1, 2, 3} is a finite set
Infinite sets
 If the number of elements of a set are not countable then it is called infinite set. or
 A set is said to be infinite if it has infinite number of elements.
For Example: A = {1, 2, 3, …….} is an infinite set.
Equivalent sets:
Two finite sets A and B are said to be equivalent sets if number of element of both the sets A and B are equal.
For example : A = {1, 2, 3, 4, 5} and B = {a, b, c, d, e}.
Number of elements of set A and B are equal but their elements are different . Hence these sets are equivalent sets.
Equal Sets
Two sets are said to be equal if
a) Number of elements of both the sets are
equal.
b) Each element of both the sets are same.
Eg. Two sets A = {1, 3, 5} and B = { 1, 1,
3, 5, 5} are said to be equal set because set B can be written as {1, 3, 5}
because in set form repeated element can be written only once.
Order or Cardinal number of a finite set :
The number of different elements of a finite set A is called the order of a set. It is denoted by n(A) or O(A).
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Set A is said to be a subset of set B if every element of set A is also the element of set B. It is denoted by A ⊂ B
A ⊂ B if a ∊ A ⇒ a ∊ B
Super Set of A
If A is the subset of B then B is called the super set of A and is denoted by B ⊃A
Proper Subset
A nonempty set A is said to be proper subset of B if A ⊂ B and A ≠ B
Φ and A are improper subsets of set A.
Comparable sets
Two sets A and B are said to be comparable sets if any one of them is the subset of other. If one set is not the subset of other then the sets are called noncomparable set.
Note
Every set is the subset of itself
Empty set (ɸ) is also the subset of every set.
N ⊂ W ⊂ Z ⊂ Q ⊂ R
Power Set: The collection of all the subsets of a set A is called the power set of A. It is denoted by P(A). Every element of P(A) is a set.
Example : Let set A = {1, 2, 3}
P(A) = {ɸ, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}
Total number of elements of P(A) = 8 = 2^{3}.
In general : If there are n elements in set A then total number of elements in the power set of A is = 2^{n}.
Universal Set : The basic set which is the super set for all the subsets used in a particular problem is called a Universal set. It is always denoted by U.
Different Types of Intervals

Intervals are the subsets of Real number (R)
Open Interval
The set of real numbers { x: a < x < b and x ∊ R } is called an open
interval and is denoted by (a, b).
All the points between a and
b belongs to the open interval (a, b) but a, b themselves do not belong to this
interval.
Close Interval
The interval
which contains the end points also is called close interval and is denoted by
[a, b], thus [a, b] = {x : a ≤ x ≤ b and x ∊ R }
We can also write intervals open at one end and close at other end.
Semiclosed(semi open) intervals
[a, b) = {x
: a ≤ x < b and x ∊ R}
(a, b] = {x
: a < x ≤ b and x ∊ R }
Graphical representations of ordered pairs
On the real
number line different intervals described above are represented as shown below.
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Most of the
relationships between sets can be represented by means of diagrams which are
known as Venn diagrams. In these diagrams universal set is denoted by rectangle
and the subsets by circles. Elements of the subsets are written in their
respective circles.
Let A and B are two sets. The union of A and B is
the set which consists of all the elements of A and all the elements of B, the
common elements being taken only once. The symbol “U” is used to denote union.
Symbolically
we write 'A U B ' and is read as ‘A union B’.
Definition of Union of sets:
The union of two sets A and B is the set which consists of all those elements
which are either in A or in B. In symbols we can write
A U B = { x :
x ∊ A or x ∊ B }
In the venn
diagram shaded portion represents A U B
Properties of union of sets
i) It holds commutative law : A U B = B U A
ii) It
holds Associative law (A U B) U
C =
A U (B U C)
iii) A U ɸ = ɸ U A = A
iv) U ⋃ A = U
The intersection of sets A and B is the set of all elements which
are common to both A and B. The symbol ‘∩’ is used to denote the intersection.
Definition of intersection of sets:
The intersection of two sets A and B is the set of all
those elements which belongs to both A and B. Symbolically we write
A ∩ B = { x : x ∊ A and x ∊ B }
Properties of intersection
i) It holds commutative law : A ∩ B = B ∩ A
ii) It
holds Associative law : A ∩ (B ∩ C) = (A ∩ B) ∩ C
iii) A ∩ ɸ = ɸ ∩ A = ɸ
vi) It holds the distributive law : A ∩ (B U C) = (A ∩ B) U (A ∩ C)
v) U ∩ A = A
Difference of sets:
The difference of two sets A and B in this
order is the set of elements which belongs to A but not to B. Symbolically, we
write A – B and read as “ A minus B” Symbolically we can write
A – B = { x : x ∊A and x ∉ B}
B – A = { x : x ∊B and x ∉ A}
Disjoint
sets:
Two sets A and B are said to be disjoint if their intersection is ‘ɸ’. Or
If A ∩ B = ɸ, then A and B are said to be disjoint sets.
Symmetric Difference of two sets:
If A and B are two sets, then the set (A  B) U (B  A) is called symmetric difference of A and B and is denoted by A Δ B.
A Δ B = {x : (x ∊ A and x ∉ B) or (x ∊ B and x ∉ A)}
Let U be a universal set and A is a subset of U. Then the
complement of A is the set of all elements of U which are not the elements of
A. complement of A is denoted by A' and is given by
A' = {x : x ∊ U and x ∉ A}
In the
following venn diagram shaded portion denotes the complement of A (or A')
Properties of complement set
i) A U A' = U ii) A ∩ A' = ɸ iii) (A')' = A
iv) ɸ' = U v) U' = ɸ
DeMorgan’s Law
i) (A U B)' = A' ∩ B' (ii)
(A ∩ B)' = A' U B'
Formulas used in practical problems
i) n(A U B) = n(A) + n(B) – n(A ∩ B)
where n represents the number of element in a set.
ii) n(A U B U C) =
n(A) +
n(B) + n(C) – n(A∩B) – n(B ∩ C) – n(C ∩ A) + n(A ∩ B ∩ C)
If A and B are two sets then
iii) n(A only) = n(A) – n(A ∩ B) = n(A  B)
iv) n(B only) = n(B)  n(A ∩ B) = n(B  A)
These formulas are also be useful for finding the probabilities (
Chapter 16)
Symbols

Meaning

U (Union)

or

∩ (Intersection)

and

 (Minus)

and not

A' (complement)

not A

At least

U (Union)

Either or

U (Union)

Neither nor

∩ (Intersection)

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