Page 34 - DMTH401_REAL ANALYSIS

Real Analysis

Notes Now, you can easily verify that all the eleven axioms are satisfied by the set of rational numbers
with respect to the ordinary addition and multiplication. Thus, the set Q forms a field under the
operations of addition and multiplication, and so does, the set R of all the real numbers.
We state (without proof) some important properties satisfied by a field. They follow from the
field axioms. Can you try?

Property 2
For any x, y, z in F,

1. x + z = y + z  x = y,
2. x, 0 = 0 = 0.x,
3. (–x). y = – x,y = x. (–y),

4. (–x), (– y) = x.y,
5. x.z = y.z, z  0 x = y,

6. x.y = 0  either x = 0 or y = 0.
Thus by now you know that the sets Q, R and C form fields under the operations of addition and
multiplication.

2.2.1 Ordered Field

We defined the order relation  in R. It is easy to see that this order relation satisfies the
following properties:

Property 3
Let x, y, z be any elements of R. Then
O : For any two elements x and y of R, one and only of the following holds:
1
(i) x < y, (ii) y < x, (iii) x = y,
O : x  y, y  x  x  z,
2
O : x  y  x  z  y + z,
3
O : x  y, 0 < z  x.z  y.z
4
We express this observation by saying that the field R is an ordered field (i.e. it satisfies the
properties 0 – 0 ). It is easy to see that these properties are also satisfied by the field Q of rational
1 4
numbers. Therefore, Q is also an ordered field. What about the field C of Complex numbers?
2.2.2 Complete Ordered Field

Although R and Q are both ordered fields, yet there is a property associated with the order
relation which is satisfied by R but not by Q. This property is known as the Order-Completeness,
introduced for the first time by Richard Dedekind. To explain this situation more precisely, we
need a few more mathematical concepts which are discussed as follows:
Consider set S = {1, 3, 5, 7). You can see that each element of S is less than or equal to 7. That is
x  7, for each x  S. Take another set S, where S = {x  R : x  17). Once again, you see that each
element of S is less than 18. That is, x < 18, for each x  S. In both the examples, the sets have
a special property namely that every element of the set is less than or equal to some number.

28 LOVELY PROFESSIONAL UNIVERSITY