
Sets 


Sets 
Definition
of a set and notation 
Set
membership 
Set
builder notation 
Cardinal
number 
Ordinal
number 
Equal
sets 
Subsets 
The
empty set or the null set 
Universal
set or universe 
Power
set 






Definition
of a Set and Notation 
Set
is a finite or infinite collection of distinct objects. Those
objects are called elements or members of the set. 
The
notation, A
= {
a, b, c
}
means A equals
the set of elements a,
b
and c. 

Set
membership 
If
B
is a set and
d
is an object, then the relationship 
d is an element of
B
we denote as
d
Î
B,
or 
d not
Î
of B if d is not an element of B,
i.e., d
does not belong to the
set B. 

Set
builder notation 
A
set can be described by
specifying the properties which determines its elements, for
example 
S
= {
n
Î
N 
3
≤
n ≤
8
} 
denotes
the set S
of all natural numbers n
such that n
is in the range from 3 to
8 inclusive that is, 
S
= {
3, 4, 5, 6, 7, 8
}. 

Cardinal
number 
A measure of the size of a set that does not take into account the order of its members, i.e., which only
specifies the total
number of its elements is called the cardinal number. 

Ordinal
number 
A
measure of a set that takes account of the order as well as the
number of its elements is called the ordinal number. 

Equal
sets 
Two sets A
and B
are said to be equal if they have the same elements. This is written A
= B. 

Subsets 
If set
S and T
are two sets such that every element of
S is also an
element of the set T,
then
S is said to be the subset
of the set T,
and is written as
S Í
T.
Every set is a subset of
itself. 

If
S is a subset of T,
but not
equal to T,
then
S is also proper subset
of T.
Therefore, a proper
subset is the one strictly contained within a larger
set, excluding some elements of the larger set. 

The
empty set or the null set 
A
set which has no members is called the empty
set or the null set and is denoted by ø
or { }. 
The
empty set is
therefore a
subset of every set. 

Universal
set or universe 
A universal set
U
is a set large enough to include all the elements of any set
relevant to a problem
under consideration.
Hence, the universal set is a set defined by the context. 

Power
set 
A
power set is a set of which the elements are all the subsets of
a given set
S including the empty
set, written P
(S)
or 2^{S}. 

Example: 
If
S
is the set { 1, 2, 3 } then power set of S,
P (S)
= { {}, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}}. 









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