11
Proof in Mathematics (cont.) — Mathematical Induction
11.1
Proof by contradiction
• §
3.6 in Epp
•
here, to prove that
p
is true, we show that assuming
∼
p
leads to a contradiction
•
classic example: prove that
√
2 is irrational
11.2
Comparing methods of proof
•
many possible approaches to proving theorems
no guarantee that any will work
sometimes many approaches work
•
an example
prove that
∀
x
∈
R
,
if
x
2

5
x
+ 4
<
0 then
x >
0
•
a direct proof
if
x
2

5
x
+ 4
<
0
then
x
2
+ 4
<
5
x
5
x > x
2
+ 4
>
0
5
x >
0
x >
0
•
an indirect proof — using contrapositive
show that
∀
x
∈
R
,
if
x
≤
0
,
then
x
2

5
x
+ 4
≥
0
if
x
≤
0
then
x

1
≤
0 and
x

4
≤
0
(
x

1)(
x

4)
≥
0
x
2

5
x
+ 4
≥
0
•
a proof by contradiction
suppose
x
2

5
x
+ 4
<
0 and
x
≤
0
then
x
2
<
5
x

4
but if
x
≤
0
5
x
≤
0
5
x

4
<
0
so
x
2
<
0
but
∀
x
∈
R
, x
2
≥
0
we have a contradiction
11.3
Existence proofs
• §
3.1 in Epp
•
useful for proving theorems of the form:
∃
x, P
(
x
)
•
constructive existence proofs
explain
e.g.
there is at least one positive integer whose value is the sum of its positive divisors
(other than itself)
6 = 1 + 2 + 3
25
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•
nonconstructive existence proofs
explain
e.g.
show that there exist irrational numbers
x
and
y
such that
x
y
is rational
consider
x
=
√
2 and
y
=
√
2
recall division into cases: (
p
∨
q
)
∧
(
p
→
r
)
∧
(
q
→
r
)
⇒
r
11.4
Proof by Counterexample
•
useful for proving theorems of the form:
∼ ∀
x, P
(
x
)
•
these theorems can be rephrased in the equivalent form:
∃
x,
∼
P
(
x
)
we only need find
one
x
for which
P
(
x
) is false
such an
x
is called a
counterexample
•
e.g.
Prove that the expression
n
2

n
+ 41 where
n
∈
Z
+
is not always prime.
show that it is prime for many values
n
= 41, however, serves as a counterexample
11.5
Mathematical induction
•
mathematical induction is another proof technique
very useful for proving theorems of the form:
∀
n
∈
Z
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 Fall '06
 Carter
 Mathematical Induction, Natural number, Mathematical logic, Mathematical proof

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