Set Theory
Set theory and its basic
foundations were developed by Georg Cantor, a mathematician from Germany
towards the end of the 19th century.
Sets
A set is a collection of
well-defined distinct objects.
The word ‘distinct’ means that the
objects of a set must be all different.
The phrase ‘well-defined’ means
that a set must have some specific property so that we can identify whether or
not an object belongs to that set.
Now, let us observe the following
collections:
1. All even natural numbers less
than 13, i.e. 2, 4, 6, 8, 10, 12.
2. Prime factors of 30, i.e. 2, 3,
5.
3. All students of your class
whose height exceeds 150 cm.
4. All months of a year.
The above collections are
well-defined collection of objects, i.e. we can say whether an object belongs
to the collection or not. So, the above collections form sets.
Now, consider the following
collections:
1. Collection of brilliant students of your class.
It is not a well-defined collection as different people might have
a different perspective on whether a student of your class is brilliant or not.
For a student getting 40% marks, a student getting 60% is brilliant while a
student getting 90% marks may call him an average student. So, it is not a set.
2. Collection of three months of a year.
It is not a well-defined collection as
it is not known which three months of a year are to be included in the
collection. So, it is not a set.
Notation of a Set
We usually denote
sets by capital letters and their elements by small letters.
All the elements
of the set are enclosed in curly brackets { } and are separated by commas (,).
Consider the set
of odd natural numbers less than 10, i.e. 1, 3, 5, 7, 9.
Let us call this
set as A. Then, A = {1, 3, 5, 7, 9}
Thus, the given set is a well-defined
collection of numbers. Also, no two numbers in it are identical.
Representation of a Set
A set can be represented
by the following methods.
i. Description method
ii. Roster method or tabular form
iii. Rule method or set builder form
1. Description Method
In this method, a
well-defined description of the elements of the set is made. The description of
elements is enclosed in curly brackets.
For example, the
set of natural numbers less than 10 is written as A = {natural numbers less
than 10}.
2. Roster Method or Tabular Form
In this method,
we list out all the elements of the set in curly brackets and separate them by
commas.
For example, the
set of 2-digit numbers whose sum of the digits is 9 is written as B = {18, 27,
36, 45, 54, 63, 72, 81, 90}.
3. Rule Method or Set Builder Form
In this method,
we write a variable representing any member of the set followed by a property,
rule or a statement satisfied by each member of the set and enclose it in curly
brackets.
For example, the
set of factors of 36 is written as C = {x : x is a factor of 36}.
Cardinal Number
of a Set
The
number of elements in a set is called the cardinal number of a set. It is
denoted by n(A).
For
example, if A = {2, 4, 6, 8, 10}, then n(A) = 5.
Types of Sets
Finite Set
If the elements
of a set can be counted, the set is called a finite set.
For example, the
set of natural satellites of Jupiter and the set of two-digit prime numbers.
Infinite Set
If the elements
of a set cannot be counted, the set is called an infinite set.
For example, the
set of fractions lying between 1 and 2 and the set of integers less than 10.
Empty Set
A set that
contains no element is called the empty set. It is denoted by the symbol { } or
ϕ.
For example, the
set of prime numbers between 5 and 7.
Singleton Set
The set which
contains only one element is called a singleton set.
For example, A =
{the number of stars in our Solar System} and B = {2}
Equal Sets
Two sets A and B
are said to be equal if they have same elements. It is written as A = B.
For example, if A
= {1, 3, 5, 7} and B = {5, 7, 3, 1}, then A = B, because the elements of A and
B are same.
Equivalent Sets
Two sets A and B
are said to be equivalent sets if they have the same number of elements,
the elements may or may not be the same. Thus, two sets A and B are equivalent
if n(A) = n(B).
For example, if A
= {colours of the rainbow}, B = {x : x is a prime number less
than 19}, then n(A) = 7 = n(B).
Overlapping Sets
Two sets A and B
are said to be overlapping sets if they have at least one element in
common.
For example, if
set A = {letters of the word DELHI} and B = {letters of the word BHOPAL} are
overlapping because the letters L and H are common in both the sets.
Disjoint Sets
Two sets A and B
are said to be disjoint sets if they have no element in common.
For example, the
set A = {x : x is a student of ABC school} and set B = {x :
x is a student of XYZ school} are disjoint sets as no student can study
in both the schools at any one point.
Complement of a Set
The complement of
a set A is the set of all elements in the universal set which are not in
set A. It is denoted by A’ and is read as ‘complement of A’.
Thus, if universal set is {1, 2, 3, 4,
5, 6, 7, 8, 9, 10} and A = {2, 4, 6, 8, 10}, then A′ = {1, 3, 5, 7, 9}.
Example 1: Express each of the following sets in the form as
required.
a. The set of
integers between –3 and 3 (Roster method)
b. C = {x :
x is a letter in the word LOLLIPOP} (Description method)
Solution:
a. {–2, –1, 0,
1, 2}
b. C = {letters of the word LOLLIPOP}
Example 2: Write the following sets in set builder form.
a. The set A
of even natural numbers lying between 5 and 20.
b. B = {10,
20, 30, 40, …}
Solution:
a. A = {x | x =
2k, 3 ≤ k ≤ 9, k ϵ N}
b. B = {x | x =
10n, n ϵ N}
Example 3: Write the following sets in roster form.
P = {x : x =
n2 + 2, n ϵ N and n ≤ 5}
Solution:
P = {3, 6, 11, 18, 27}
Example 4: Find the cardinal number of the following sets.
a. A = {x
: x
= n2 + 1, n
ϵ N and n ≤ 4}
b. B = {x
: x
is a day
of a week}
c. D = {x
: x
is a
letter of the word ‘NATIONAL’}
Solution:
a. Here,
A = {2, 5, 10, 17} which has 4 elements.
Hence, n
(A) = 4
b. Here,
B = {Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}
Hence,
n
(B) = 7
c. Here,
D = {N, A, T, I, O, L}
Hence,
n
(D) = 6
Example
5: State
whether each of the following statements is true or false when
A =
{letters of the word ‘NUMERAL’} and B = {letters of the word ‘MATERIAL’}.
a. A and
B are equal sets.
b. A and B are equivalent sets.
c. A and
B are disjoint sets.
Solution:
Here, A
= {N, U, M, E, R, A, L} and B = {M, A, T, E, R, I, L}
a. False
as the elements are not identical
b. True as n (A) = n
(B) = 7
c. False as elements M, E, R, L and A are in both the sets.
