Page 12 - DMTH403_ABSTRACT_ALGEBRA

Unit 1: Generating Sets

Let A and B be two sets. Consider the pair (a, b), in which the first element is from A and the Notes
second from B. Then (a, b) is called an ordered pair. In an ordered pair the order in which the two
elements are written is important. Thus, (a, b) and (b,a) are different ordered pairs. Two ordered
pairs (a, b) and (c, d) are called equal, or the same, if
a = c and b = d.
Definition: The Cartesian product A x B, of the sets A and B, is the set of all possible ordered pairs
(a, b), where a  A, b  B.
For example, if A = {l , 2 , 3} and B = (4, 6), then
A × B = { (1, 4), (1, 6), (2, 4), (2, 6), (3, 4), (3, 6) }.
Also note that

B × A = { (4, 1), (4, 2), (4, 3), (6, 1), (6, 2), (6, 3) } and A x B  B x A.
Let us make some remarks about the Cartesian product here.
Remarks: (i) A x B =  iff A =  or B = .
(ii) If A has m elements and B has n elements, then A x B has mn elements. B x A also has mn
elements. But the elements of B x A need not be the same as the elements of A x B, as you have just
seen.

We can also define the Cartesian product of more than two sets in a similar way. Thus, if A , A ,
1
2
A , .......... A , are n sets, we can define their Cartesian product as
n
3
A x A x .......x A = { (a , a ,......., a ) | a  A ,.........., a  A }.
n
1
2
n
n
1
l
n
l
2
For example, if R is the set of all real numbers, then
R x R = { (a , a ) | a  R, a  R }
2
2
l
1
R x R x R = { (a , a , a ) | a  R for i = 1, 2, 3 ), and so on. It is customary to write R for R x R and
2
l
2
3
i
R for R x .......... x R (n times).
n
Now, you know that every point in a plane has two coordinates, x and y. Also, every ordered
pair (x, y) of real numbers defines the coordinates of a point in the plane. So, we can say that R 2
represents a plane. In fact, R is the Cartesian product of the x-axis and the y-axis. In the same way
2
R represents three-dimensional space, and R represents n-dimensional space, for any n  1.
n
3
Note that R represents a line.
1.3 Relations
You are already familiar with the concept of a relationship between people. For example, a
parent-child relationship exists between A and B if and only if A is a parent of B or B is a parent
of A.
In mathematics, a relation R on a set S is a relationship between the elements of S. If a  S is
related to b  S by means of this relation, we write a R b or (a, b)  R. From the latter notation we
see that R  S × S. And this is exactly how we define a relation on a set.
Definition: A relation R on a set S is a subset of S × S.
For example, if N is the set of natural numbers and R is the relation ‘is a multiple of’, then
15 R 5, but not 5 R 15. That is, (15, 5)  R but (5, 15)  R. Here R  N × N.
Again, if Q is the set of all rational numbers and R is the relation ‘is greater than’, then 3 R 2
(because 3 > 2).
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