Definition (Union): The union of sets
A and B, denoted by A
B , is the set defined as
A
B = {
x | x
A
x
B }
Example 1: If A = {1, 2, 3}
and B = {4, 5} , then A
B = {1, 2, 3, 4, 5} .
Example 2: If A = {1, 2, 3} and B
= {1, 2, 4, 5} , then A
B = {1, 2, 3, 4, 5} .
Note that elements are not repeated in a set.
Definition (Intersection): The intersection of sets A and
B, denoted by A
B , is the set defined as
A
B = {
x | x
A
x
B }
Example 3: If A = {1,
2, 3} and B = {1, 2, 4, 5} , then
A
B =
{1, 2} .
Example 4: If A = {1, 2,
3} and B = {4, 5} , then A
B =
.
Definition (Difference): The difference of sets A from B
, denoted by A - B (or A \ B), is the set defined as
A - B = { x | x
A
x
B }
Example 5: If A = {1, 2, 3} and
B = {1, 2, 4, 5} , then A - B
= {3} .
Example 6: If A = {1, 2, 3}
and B = {4, 5} , then A - B =
{1, 2, 3} .
Note that in general A -
B
B - A
Definition (ordered pair): An ordered
pair is a pair of objects with an order associated with them.
If
objects are represented by x and y, then we write
the ordered pair as (x, y).
Two ordered pairs
(a, b) and (c, d) are equal if and only if a
= c and b = d. For example the ordered
pair (1, 2) is not equal to the ordered pair (2,
1).
Definition (Cartesian product):
The set of all ordered pairs (a, b), where
a is an element of A and b is an
element of B, is called the Cartesian
product of A and B and is denoted by
A
B.
Example 1: Let
A = {1, 2, 3} and B = {a, b}. Then
A
B = {(1, a),
(1, b), (2, a), (2,
b), (3, a), (3, b)} .
Example 2: For the same A and B as in
Example 1,
B
A = {(a, 1), (a, 2),
(a, 3), (b, 1), (b,
2), (b, 3)} .
As you can see in these
examples, in general, A
B
B
A unless A =
,
B =
or A = B.
Note
that A
=
A =
because
there is no element in
to form ordered pairs with elements of
A.
The concept of Cartesian product can be extended to
that of more than two sets. First we are going to define the concept of
ordered n-tuple.
Definition (ordered n-tuple): An ordered n-tuple is a set of n
objects with an order associated with them . If n objects are represented by x1,
x2, ..., xn, then we write the
ordered n-tuple as (x1,
x2, ..., xn) .
Definition (Cartesian product): Let A1, ...,
An be n sets. Then the set of all
ordered n-tuples (x1, ...,
xn) , where xi
Ai for all i, 1
i
n
, is called the Cartesian product of
A1, ..., An, and is denoted by
A1
...
An .
Example 3:
Let A =
{1, 2}, B = {a, b} and C = {5, 6}. Then
A
B
C = {(1, a, 5), (1, a, 6),
(1, b, 5), (1, b, 6), (2, a,
5), (2, a, 6), (2, b, 5),
(2, b, 6)} .
Definition (equality of n-tuples): Two
ordered n-tuples (x1, ...,
xn) and (y1, ...,
yn) are equal if
and only if xi = yi for
all i, 1
i
n
.
For example the ordered 3-tuple (1, 2,
3) is not equal to the ordered n-tuple (2, 3,
1).
Definition
(binary relation):
A binary
relation from a set A to a set B is a
set of ordered pairs (a, b) where a is an
element of A and b is an element of B.
When an ordered pair (a, b) is in a relation
R, we write a R b, or (a, b)
R. It means that element a is related
to element b in relation R.
When A =
B, we call a relation from A to B a
(binary) relation on A .
Examples:
If A = {1, 2, 3} and
B = {4, 5}, then {(1, 4), (2, 5), (3,
5)}, for example, is a binary relation from A to
B.
However, {(1, 1), (1, 4), (3,
5)} is not a binary relation from A to B
because 1 is not in B.
The parent-child relation
is a binary relation on the set of people. (John, John Jr.),
for example, is an element of the parent-child relation if John is the father of
John Jr.