A
⊂
B
The null set or empty set (
∅
) includes no elements.
The union of two sets is denoted by
A
∪
B
, is the set of all
elements contained in
A
, in
B
or in both
A
and
B
.
A
∪
B
={
x

x
∈
A
or
x
∈
B
}
Example: A = {3, 5, 7} B = {2, 3, 4, 8}
A
∪
B
= {2, 3, 4, 5, 7, 8}
The intersection of two sets is denoted by
A
∩
B
, is the set
of all elements contained in both A and B.
A
∩
B
={
x

x
∈
A
or
x
∈
B
}
Example: A = {3, 5, 7} B = {2, 3, 4, 8}
A
∩
B
= {
3
}
If two set have no common elements, they are called
disjoint sets
. Set A and B are disjoint if
A
∩
B
=
∅
.
In general, the order of the set does not matter. In sets of
certain types, though, order maters.
These are called
ordered pairs and written as (a, b).
The
Cartesian product
of two sets,
x
×
y
, is the set of all
ordered pairs, where the first element comes from set
x
and the second element comes from set
y
.
x
×
y
= {(
a, b
)

a
∈
x
and
b
∈
y
}
Example 1:
Euclidean 2space is
ℜ
×
ℜ
=
ℜ
2
Example 2:
x = {1,2} and y = {3,4}
x
×
y
= {(1,3)(1,4)(2,3)(2,4)}
Complement of set
A
is .
2.
Relations and Functions: