Page 15 - DMTH502_LINEAR_ALGEBRA

Unit 1: Vector Space over Fields

This is known as the well ordering property of natural numbers. Notes
The relations between order and addition, and order and multiplication are given by the following
results:

(i) m > n m + p > n + p,
(ii) m > n m p > n p, for all m, n, p N.
The operation of subtracting a number n from another number m is possible only when m > n, i.e.,
the subtraction operation is not defined for any two natural numbers. It is thus not a binary
composition in N.
Similarly the operation of dividing one number is also not always possible, i.e., the division
operation is also not a binary composition in N.

Integers

The set of integers is constructed from the set of natural numbers by defining a relation, denoted
by “~” (read as wave), in N × N as follows:

( , )~( , ) if a d b c , , , ,d N .
d
c
b
b
a
a
c
Since this relation is an equivalence relation it decomposes the set N × N into disjoint equivalence
classes. We define the set of all these equivalence classes as the set of integers and denote it by Z.
The equivalence class of the pair (a, b) may be denoted by
(a, b) or (a, b)*
The addition and multiplication operations in Z are now defined as follows:
(a, b)* + (c, d)* = (a + c, b + d)*
and (a, b)* . (c, d)* = (ac + bd, ad + bc)*.
The associative and commutative laws of addition and multiplication hold as for natural numbers. The
cancellation law of addition holds in general, but the cancellation law of multiplication holds with
some restrictions. The distributive law of multiplication over addition is also valid.
The equivalence class (1, 1)* is defined as the integer zero, and is written as 0. Thus
0 = (1, 1)* = (a, a)* = (b, b)*, a, b N.
This number 0 possesses the properties, that for any integer x,
(i) x + 0 = x and

(ii) x . 0 = 0.
If x = ( , )* is an integer other than zero, we have , i.e., either > or < . We say that the
integer ( , )* is positive if > and negative if < .

When > , , N, there exists a natural number u such that = u + .
Therefore a positive integer x is given by

x ( , )*, , (u , )* (u , )* .
It is possible to identify the positive integer (u + , )* with the natural number u, and write it
as + u. Thus the set of positive integers may be written as
Z = {+1, +2, +3, …}
N

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