Download When to Use Indirect Proof

When to Use Indirect Proof
Many theorems can be proved either way. Usually, however,
when both types of proof are possible, indirect proof is clumsier
than direct proof.
In the absence of obvious clues suggesting indirect argument, try
first to prove a statement directly. Then, if that does not
succeed, look for a counterexample.
If the search for a counterexample is unsuccessful, look for a
proof by contradiction
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If
number
a
Y
h
if
is
hi
the
then
number
number
is
th
be
h
uhrtrary
an
'
.
odd
his
.
Suppose there
then
IKH
B
h
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he
definition
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for
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odd
is
ht
ns.t
even
of
Meger
some
.
h
'
is
By def of
.
.
integers
.
odd
K
.
odd
Butbut
even
n
h
-
2k
Some
h 't
odd
=P
T
But
odd
on
then
odd
is
squared
( IK ) ?_
YK
'
=
2(
IKT
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hrs
even
A
contradiction
for
int .K
.
.
'
If
is
h
fn
if
then
even
li
is
his
then
even
Suppose
n
is
✓
.
even
=
=
Proof ( by
even
)
contradiction
that
there
is
ns.t
.
mm
h#bAh
his
is
then
But
h=2k←l
then
A
.
ht
So
odd
( LKHHYK
definition
contradiction
.
't
4×+1=2126*4
of odd
The Real Numbers
-
All (decimal) numbers — distances to points on a number line.
Examples.
to
−3.0
0
°
1.6
1
π = 3.14159 . . .
-
A real number that is not rational is called irrational.
.
But are there any irrational numbers?
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?
} Ren#
Fnirgdnumd
ff
is
Pr#
not
rational
a
for
I
Then
BE
that
suppose
Additionally
number
assume
rational
integers
some
that
,←q#
(
Then
'
?_ ni
2h
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era
M§forsownkgerT→
'
m
is
even
So
Nisan
.
.
Acowtydton
.
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U
2n2=(2lH4t
Then
m
hto
st
mm
.n=m
fnpem
1
F
.
:
We
fraction
So
assumed
so
mad
HER
that
h
'
th
cannot
.
.
is
be
even
a
at
reduced
the
save
time
Proving that
√
2 is not a rational number
O
Proof by contradiction.
√
√
If 2 were rational then we could write it as 2 = x/y where
x and y are integers and y is not 0.
By repeatedly cancelling common factors, we can make sure that
x and y have no common factors so they are not both even.
Then 2 = x2 /y 2 so x2 = 2y 2 so x2 is even. This means x is
even, because the square of any odd number is odd.
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the proof continued
Let x = 2w for some integer w.
Then x2 = 4w2 so 4w2 = 2y 2 so y 2 = 2w2 so y 2 is even so y is
even.
This contradicts the fact that x and y are not both even, so our
√
original assumption, that 2 is rational, must have been wrong.
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Part 1. Number Systems and Proof Techniques
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√
Prove that 1 + 3 2 is irrational
Proof ( by
that
Support
1+3 K
r=
contradiction
Then
#
.
Then
Then
A
K
r
]
KB
rational
.
1=352
.
K
is
contradiction
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It
F-
)
rational
a
hunw
.
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