Set operations
We now use connectives to define the
set operations[1], these allow is to build new set from given ones. Let
and
be subsets of the set
.
Definition : The union
The union of
and
is
is the set of all elements that are in
or in
.

Definition : The intersection
The intersection of
and
is
is the set of all elements that are in both
and
.

Example :
If
and
then
and
Method :
Let
and
be finite sets, then we have
.
Definition :
The sets
and
disjoint when
.
Definition : The set difference
The set difference of
and
, or relative complement of
with respect to
, written
and read “
minus
” or “the complement of
with respect to
,” is the set of all elements in
that are not in
.
.

Example :
,
and
then
,
: irrational numbers.
Definition : The complement
The complement of the set
, written
and read “the complement of
,” is the set of all elements of
that are not in
. That is,
.

Some properties
From the proprieties of the logical operations we drive the following,
Method :
Cartesian products
The cartesian product of two or more sets is the set of all ordered pairs/n-tuples of the sets. It is most commonly implemented in set theory. Cartesian product is the product of any two sets, but this product is actually ordered i.e, the resultant set contains all possible and ordered pairs such that the first element of the pair belongs to the first set and the second element belongs to the second set. Since their order of appearance is important, we call them first and second elements, respectively.
Definition :
The Cartesian product of
and
is
Example :
is called the Cartesian plane.
If
and
then
.
Note :
and
.
Sub sets
Subsets are a core concept in the study of Set Theory[2], similar to Sets.
Definition :
A set
is a subset of another set
if all elements of the set
are elements of the set
. We write
. If
is not a subset of
, we write
.
Example :
.
Method :
Let
be sets, then
.
.
If
and
.
Families of sets
The elements of a set may themselves be sets. For example, the power set of a set
,
is the set of all subsets of
. The phrase, “a set of sets” sounds confusing, and so we often use the terms collection and family when we wish to emphasize that the elements of a given set are themselves sets. We would then say that the power set of
is the family (or collection) of sets that are subsets of
.
Definition :
For any set
,
.
Example :
The set
then