11
Indirect proof: Proof by contrapositive.
(
A
=
⇒
B
)
≡
(
¬
B
=
⇒ ¬
A
)
,
so show
¬
B
=
⇒ ¬
A
directly.
Example 0.2.3
(contrapositive)
.
3
n
+ 2 odd =
⇒
n
odd.
The contrapositive is:
n
even =
⇒
3
n
+ 2 even.
1. Assume
n
is an even integer.
2. Then
n
= 2
k
, for some integer
k
, so
3
n
+ 2 = 3(2
k
) + 2 = 6
k
+ 2 = 2(3
k
+ 1) = 2
m,
for some
m
∈
Z
.
3. Thus, 3
n
+ 2 is even.
Example 0.2.4
(contrapositive)
.
n
2
even =
⇒
n
even.
This is just the contrapositive of the prev. example.
Indirect proof: Proof by contradiction.
In order to show that
A
is true by contradiction,
1. assume that
A
is false (assume
¬
A
is true)
2. derive a contradiction (show that
¬
A
implies something which is clearly false/impossible)
Example 0.2.5
(contradiction)
.
√
2 is irrational.
1. Assume the negative of the statement:
√
2 =
m
n
, for some
m,n
∈
Z
.
2. If
m,n
have a common factor, we can cancel it out to obtain
√
2 =
a
b
,
in lowest terms
(
*
)
2 =
a
2
b
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 Fall '11
 Wong
 Logic, Indirect Proof, Mathematical Induction, Natural number, Mathematical logic, Greatest common divisor

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