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In-Depth Information
11
(i) proposition
justification
a
b
given
finition
b
a
is positivede
0
c
given
finition
c
is positivede
(
b
r
a
) ·
c
is positiveclosure
unde
multiplication
b
·
c
a
·
c
is positivealge
bra
a
·
c
b
·
c
definition
(ii) Put
b
0.
(iii)
ac
bc
bd
.
12
proposition
justification
a
b
given
finition
a
·
c
b
·
c
b
a
positivede
given
0 hypothesis
a
·
c
b
·
c
algebra
This contradicts what is given so hypothesis is false.
c
c
0
hypothesis
finition
0
c
is positivede
(
b
a
) · (
c
) is positiveclosure
r multiplication
unde
a
·
c
b
·
c
is positivealge
bra
a
·
c
definition
This contradicts what is given so hypothesis is false.
b
·
c
0
c
trichotomy and qn 11
Sherlock Holmes said that when what is impossible has been
eliminated, what remains, however improbable, must be the truth.
The only difference between the 'Sherlock Holmes method' and a
mathematical proof by contradiction, is that in mathematics the
final conclusion is what we were really expecting all along.
14
2
1, but 1
(
2)
4.
proposition
justification
a
b
and 0
a
given
a
ab
qn 11
a
b
and 0
b
given, transitivity
ab
b
qn 11
a
ab
and
ab
b
above
a
b
transitivity
