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with them and what you need to be careful about when you are using
them in an argument.
1 To prepare for a formal treatment of inequalities, determine for
what numbers
x
you want to claim theinequality 2
x
3
x
. Check
whether you expect it to hold when
x
1, when
x
0 and when
x
1.
1. If
a
is a number, then
either a
0,
or a
is positive
or
a
is
positive, and only one of these is true. When
a
is positive,
a
is said to be negative. This property is called the
trichotomy
law
because of the
three
possibilities.
2. The sum of two positive numbers is positive. This is also
described by saying that the positive numbers are
closed under
addition
.
3. The product of two positive numbers is positive. This is also
described by saying that the positive numbers are
closed under
multiplication
.
Weintroducetheinequality '
' with thefollowing
' '
Wesay
a
b
if and only if
b
a
is positive.
The subset of positive numbers in Z is denoted by Z
, thesubst of
positivenumbrs in Q is denoted by Q
, and thesubst of positive
numbers in R is denoted by R
.
We can go on from here to define
b
a
if and only if
a
b
,
and to modify these definitions for
and
. We will keep to '
'
until some elementary properties have been established. In the
proofs, be careful only to use the properties of '
' which have been
given, or which you have successfully deduced from them. The
answers to these questions have been set out so as to highlight the
reasons for each step.
2 Usethe
definition of less than
to give an inequality equivalent to the
proposition '
b
is positive'.
3 If 0
a
, usethedefinition to provethat
a
0.
4 If
a
0, usethedefinition to provethat 0
a
.
