
Operations
on Sets and Venn Diagrams 


Operations on sets and Venn diagrams 
Union,
intersection and difference 
Partition
of a set 





Operations
on Sets and Venn Diagrams 
There are three basic set operations:
union, intersection and difference (relative complement).

The union of two sets A
U B
is the set that consists of all elements belonging to either
set A
or set B
(or both).
We
say that an element x
is in A
U B
if either x
is in A
or x
is in B (or
x
belongs
to both). 
In
formal notation, A
U B
= {
x 
x Î
A or
x Î
B or
both }. 
The intersection of two sets A
∩ B
is the set of all elements that are elements of both A
and B. 
In
formal notation, A
∩ B
= {
x 
x Î
A and
x Î
B }. 


The union and the intersections of sets A
and B
represented by Venn
Diagrams. 

The difference
(or the relative complement of B
in A),
denoted as A − B
(or A
\ B),
is the set of all elements
that are elements of A
but not of B.
In formal notation, A
− B
= {
x 
x Î
A and
x not
Î
of B }. 
The
complement (or the absolute complement) of A
in U,
denoted as A'
(or A^{c}),
is the set of all elements of a
given universal set that are not elements of A.
In formal notation, A'
= {
x 
x Î
U
and
x not
Î
of A
}. 


The difference A
− B
and the complement A'
represented by Venn
Diagrams. 

Example: 
Assuming
the set of natural numbers N
is the universal set given are sets, 

A = {
x Î
N

x <
5
},
B
= {
x Î
N

3
≤ x <
8 }
and
C = {
x Î
N

x =
2n
− 1,
n
Î N
}. 
Find: a) A
∩ B,
b) A
U B,
c)
A
− B,
d)
B
− A
and e)
C'. 

Solution:
As A
= {
x Î
N

x <
5
} =
{ 1, 2, 3, 4 }, 
B = {
x Î
N

3
≤ x <
8 }
=
{ 3, 4, 5, 6, 7 } 
and C
= {
x Î
N

x =
2n
− 1,
n
Î
N
}
= {
1, 3, 5, 7, . . . } 
then a) A
∩ B
=
{ 3, 4 },
b) A
U B
=
{ 1, 2, 3, 4, 5, 6,
7 }, 
c)
A
− B
=
{ 1, 2 },
d)
B
− A
=
{ 5, 6, 7 } 
and
e) C'
= {
x Î
N

x =
2n
} = {
2, 4, 6, 8, . . . }. 

Partition
of a Set 
A partition of a set
S is a collection of nonempty disjoint subsets of
S, whose union is
S. Two sets are said to be
disjoint if they have no elements in common. 

Example: 
If
given set is {1, 2, 3, 4, 5, 6, 7} then one of its
possible partitions is { {1, 3, 7}, {2}, {4, 5}, {6} }. 









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