Graphics Reference
In-Depth Information
5 Usethelaw of trichotomy to provethat for any numbr
a
, either
a
0or0
a
or
a
0, and only one of these is true.
6 If
b
a
, usethedefinition to provethat
a
b
.
7
Extended trichotomy law
Apply thetrichotomy law to thenumbr
b
a
to provethat for
any two numbers
a
and
b
,
either a
b
,
or a
b
,
or b
a
, and
exactly one of these is true.
8 If
a
b
, use the definition, and closure under addition, to prove
a
c
b
c
.
9
Transitive law
If
a
b
and
b
c
, provethat
a
c
.
When both
a
b
and
b
c
, weusually write
a
b
c
.
It is thecombination of
extended trichotomy
with the
transitive law
that makes it so helpful to mark numbers, in order, along a straight
line.
10 If
a
b
and
c
d
, provethat
a
c
b
d
.
11
(i) If
a
b
and 0
c
, provethat
a
·
c
b
·
c
.
(ii) Deduce that if
a
0 and 0
c
, then
a
·
c
0.
(iii) If 0
a
b
and 0
c
d
, provethat
ac
bd
.
12 If
a
b
and
a
·
c
b
·
c
, provethat 0
c
.
This is closely related to question 11(i). But the constructive
methods we have used so far do not provide a proof. If you
supposethat theconclusion is
wrong
, you can contradict what has
been given. The contradiction is the means of showing that in fact
theconclusion must beright. Such a proof is called a
proof by
contradiction
.
13 Sketch a graph of the line
y
cx
for positive
c
to illustratethe
consequent relationship between
a
b
and
ca
cb
, as in questions
11 and 12. Sketch a graph of the line
y
cx
for negative
c
to
illustrate the consequent relationship between
a
b
and
cb
ca
.
14 Givean exampleto show that it is possibleto have
b
a
when
a
b
.If0
a
b
, provethat
a
b
.
15 If
a
0, provethat 0
a
.
16 Useqn 15 to provethat 0
1. Deduce that there is no number
a
such that
a
1 in a number system with inequalities.
