Induction and Recursion

Definitions
o
sequence:
a function with domain that is a subset of the set of integers
o
geometric progression:
a sequence of the form
,
,
,
2
ar
ar
a
, where a and r are real numbers
o
arithmetic progression:
a sequence of the form
,
2
,
,
d
a
d
a
a
+
+
, where a and d are real
numbers
o
the principle of mathematical induction:
The statement
)
(
n
nP
2200
is true if
)
1
(
P
is true and
[
]
)
1
(
)
(
+
→
2200
k
P
k
P
k
is true
o
basis step:
the proof of
)
1
(
P
in a proof by mathematical induction of
)
(
n
nP
2200
o
inductive step:
the proof of
)
1
(
)
(
+
→
k
P
k
P
for all positive integers k in a proof by
mathematical induction of
)
(
n
nP
2200
o
strong induction:
The statement
)
(
n
nP
2200
is true if
)
1
(
P
and
(
29
[
]
)
1
(
)
(
^
)^
1
(
+
→
2200
k
P
k
P
P
k
is true
o
wellordering property:
Every nonempty set of nonnegative integers has a least element
o
recursive definition of a function:
a definition of a function that specifies an initial set of values
and a rules for obtaining values of this function at integers from its values at smaller integers.
o
recursive definition of a set:
a definition of a set that specifies an initial set of elements in the set
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 Fall '07
 SELMAN
 Mathematical Induction, Recursion, Natural number, set structural induction

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