L4.12
Lecture Notes: Mathematical Induction
It is often the case that we want to consider the truth of statements such as
(1)
∑
=
n
i
0
i = n(n+1)/2,
n
≥
0
This means that the sum from zero to
any
natural number is given by the
formula. This is another instance of a hidden universal quantifier. We could
state it more explicitly by saying
for all natural numbers,
∑
=
n
i
0
i = n(n+1)/2
or
2200
n
∑
=
n
i
0
i = n(n+1)/2
(1)
is clearly a proposition. It is either true or false. The variable
n
is bound
by the universal quantifier. Let’s symbolize the formula in logic.
Let P(n):
∑
=
n
i
0
i = n(n+1)/2
P is a predicate and P(n) is a propositional function just like the ones we
studied in logic. Give it a value for
n
, and it gives you back a truth value, for
example:
P(5):
∑
=
5
0
i
i = 5(5+1)/2 = 15 is True since 0+1+2+3+4+5=15.
In a universe of natural numbers, we may restate (1) as
2200
nP(n).
It is often the case that we want to prove such statements as (1), but so far
we have no way of doing so. Proving (1) is difficult because an infinite
number of formulas have to be proved, namely for n=0, 1, . .
It turns out that there is a rule of inference
that will do the job for us. We
will prove that rule of inference below. For now you may accept the
following rule of inference as valid.
1
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View Full DocumentRULE OF INFERENCE
.
P(0)
Premise
2200
n(P(n)
→
P(n+1))
Premise
—————————
2200
nP(n)
Conclusion
If we can show the truth of the two premises (somehow), then we are
allowed to say we have proved the conclusion. This rule of inference has the
name
mathematical induction
.
This is consistent with our habit of naming
rules of inference that we use often, like
modus ponens
or
universal
instantiation
. Proofs that use mathematical induction as a rule of inference
are often called “inductive proofs.”
Intuitively, if we prove P for the value 0, then the truth of the second
premise says that P is true for the value 1. But if it is true for 1, the second
premise says that it is true for 2, and so ad infinitum.
P(0) is proved directly.
This step is the
basis.
The second premise is proved by using universal instantiation, and then
using the proof method for implications, namely assuming the antecedent
and proving the consequent. When we combine these we get a statement like
this:
Assume that P(n) is true for an arbitrary n. Based on this assumption
, show
that P(n+1) is true.
(At this point the reader is advised to review the proof methods for
implications given in Lecture Notes 1.57, p 8)
(The curious reader may ask why we don’t simply attempt to prove the
conclusion
2200
nP(n) by proving P for an arbitrary
n
. The problem is we don’t
know how to express an arbitrary natural number by itself. But in an
implication we can work with an arbitrary
n
in relation to
n+1
. That’s an
easier concept to work with, and that’s the strength of this rule of inference.
It’s important to note that the
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 Spring '08
 WATKINS
 Natural Numbers, Mathematical Induction, Natural number, Mathematical proof

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