Introduction to Sets

INDEX
• SETS
• TYPES OF SETS
• OPERATION ON SETS

SET
• A set is a well defined collection of objects, called the “elements” or “members” of the
set.
• A specific set can be defined in two ways-
1. If there are only a few elements, they can be listed individually, by writing them between curly
braces ‘{ }’ and placing commas in between. E.g.- {1, 2, 3, 4, 5}
2. The second way of writing set is to use a property that defines elements of the set.
e.g.- {x | x is odd and 0 < x < 100}
• If x is an element o set A, it can be written as ‘x  A’
• If x is not an element of A, it can be written as ‘x  A’

SPECIAL SETS
• Standard notations used to define some sets:
a. N- set of all natural numbers
b. Z- set of all integers
c. Q- set of all rational numbers
d. R- set of all real numbers
e. C- set of all complex numbers

SUBSET
• If every element of a set A is also an element of set B, we say set A is a subset of set B.
A  B
Example-
If A={1,2,3,4,5,6} and B={1,2,3,4}
Then B  A

EQUAL SETS
• Two sets A and B are called equal if they have equal numbers and similar types of
elements.
i.e. A  B and B  A .
This implies, A=B
• For e.g. If A={1, 3, 4, 5, 6}
B={4, 1, 5, 6, 3} then both Set A and B are equal.

EMPTY SETS
• A set which does not contain any elements is called as Empty set or Null or Void set. Denoted by 
or { }
• example: (a) The set of whole numbers less than 0.
(b) Clearly there is no whole number less than 0. Therefore, it is an empty set.
(c) N = {x : x ∈ N, 3 < x < 4}
• Let A = {x : 2 < x < 3, x is a natural number}
Here A is an empty set because there is no natural number between 2 and 3.
• Let B = {x : x is a composite number less than 4}.
Here B is an empty set because there is no composite number less than 4.

SINGLETON SET
• A singleton set is a set containing exactly one element.
• Example: Let B = {x : x is a even prime number}
Here B is a singleton set because there is only one prime number which is even, i.e., 2.
• A = {x : x is neither prime nor composite}
It is a singleton set containing one element, i.e., 1.

FINITE SET
• A set which contains a definite number of elements is called a finite set. Empty set is also
called a finite set.
For example:
• The set of all colors in the rainbow.
• N = {x : x ∈ N, x < 7}
• P = {2, 3, 5, 7, 11, 13, 17, ...... 97}

INFINITE SET
• The set whose elements cannot be listed, i.e., set containing never-ending elements is called an
infinite set.
For example:
• Set of all points in a plane A = {x : x ∈ N, x > 1}
• Set of all prime numbers B = {x : x ∈ W, x = 2n}
Note:
• All infinite sets cannot be expressed in roster form.

CARDINAL NUMBER OF A SET
• The number of distinct elements in a given set A is called the cardinal number of A. It is denoted by
n(A).
• For example:
A {x : x ∈ N, x < 5}
A = {1, 2, 3, 4}
Therefore, n(A) = 4
B = set of letters in the word ALGEBRA
B = {A, L, G, E, B, R} Therefore, n(B) = 6

DISJOINT SETS
• Two sets A and B are said to be disjoint, if they do not have any element in common.
• For example:
A = {x : x is a prime number}
B = {x : x is a composite number}.
Clearly, A and B do not have any element in common and are disjoint sets.

POWER SET
• The collection of all subsets of set A is called the power set of A. It is denoted by P(A).
In P(A), every element is a set.
• For example;
If A = {p, q} then all the subsets of A will be
P(A) = {∅, {p}, {q}, {p, q}}
Number of elements of P(A) = n[P(A)] = 4 = 22
In general, n[P(A)] = 2m where m is the number of elements in set A.

UNIVERSAL SET
• A set which contains all the elements of other given sets is called a universal set. The symbol
for denoting a universal set is ∪ or ξ.
• For example;
1. If A = {1, 2, 3} B = {2, 3, 4} C = {3, 5, 7}
then U = {1, 2, 3, 4, 5, 7}
[Here A ⊆ U, B ⊆ U, C ⊆ U and U ⊇ A, U ⊇ B, U ⊇ C]
2. If P is a set of all whole numbers and Q is a set of all negative numbers then the universal set
is a set of all integers.
3. If A = {a, b, c} B = {d, e} C = {f, g, h, i}
then U = {a, b, c, d, e, f, g, h, i} can be taken as universal set.

• The four basic operations are:
• 1. Union of Sets
• 2. Intersection of sets
• 3. Complement of the Set
• 4. Cartesian Product of sets

UNION OF SET
• Union of two given sets is the smallest set which contains all the elements of both the
sets.
A  B = {x | x  A or x  B}
A B

INTERSECTION SET
• Let a and b are sets, the intersection of two sets A and B, denoted by A  B is the set
consisting of elements which are in A as well as in B
• A  B = {X | x  A and x  B}
• If A  B= , the sets are said to be disjoint.
A B
A  B

COMPLEMENT OF A SET
• If U is a universal set containing set A, then U-A is called complement of a set.
A

Introduction to Sets

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