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A proof by mathematical
induction (POMI):
To prove “
P
(
n
) is true for all
n
≥
1”:
Base Case
:
Check
P
(1).
Induction Hypothesis
:
Assume that
P
(
k
) is true for some
k
≥
1.
Induction Conclusion
:
Prove that
P
(
k
+ 1) is true using Ind. Hyp.
1
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View Full Document Example.
2
n
>
4
n
for all
n
≥
5.
Base Case
:
n
= 5 :
2
5
= 32
>
20 = 4
·
5.
Induction Hypothesis
:
Assume that 2
k
>
4
k
for some
k
≥
5.
Induction Conclusion
:
We prove 2
k
+1
>
4(
k
+ 1)
2
k
+1
= 2
k
+ 2
k
≥
(since
k
≥
5
≥
2)
2
k
+ 2
2
>
(by Ind.Hyp.
)
4
k
+ 4 = 4(
k
+ 1)
.
2
A proof by strong
induction (POSI):
To prove “
P
(
n
) is true for all
n
≥
1”:
Base Case
:
Check
P
(1).
Induction Hypothesis
:
Assume
P
(1)
, P
(2)
, . . . , P
(
k
) for some
k
≥
1.
Induction Conclusion
:
Prove that
P
(
k
+ 1) is true using Ind. Hyp.
3
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View Full Document To prove “
P
(
n
) is true for all
n
≥
1”:
Check
P
(1)
, P
(2)
, . . . P
(
m

1)
,
and prove by induction
“
P
(
n
) is true for all
n
≥
m
”
Base Case
:
Check
P
(
m
).
Induction Hypothesis
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This note was uploaded on 02/05/2010 for the course MATH 135 taught by Professor Andrewchilds during the Spring '08 term at Waterloo.
 Spring '08
 ANDREWCHILDS
 Mathematical Induction

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