Elementary Proof Techniques
These methods are used when trying to prove that the conditional
P
=
)
Q
is true.
Vacuous proof:
prove that
P
=
)
Q
is true by showing that
P
is
always false. This makes the conditional always true, regardless of the
truth value of the conclusion
Q
. Recall that that the truth value of a
conditional is not the same as showing that the conclusion is true.
Trivial proof:
prove that
P
=
)
Q
is true by showing that
Q
is true
without using that
P
is true.
Direct Method:
prove
P
=
)
Q
by starting with the assumptions in
P
and by following logical deductions, obtain
Q
Contrapositive Method:
uses the fact that
P
=
)
Q
is logically
equivalent to
q
Q
=
)
q
P
. Sometimes, if
Q
is a negation itself, it is
easier to start by assuming
q
Q
is true.
Method by contradiction:
Assumes that
P
and
q
Q
are true, obtain
a contradiction. Or, if the statement is "Show
P
is true", we can
assume
q
P
is true and reach a contradiction.
Remark 1
as follows: "For all
n
2
N
,
P
(
n
)
is true" can be written as
n
2
N
=
)
P
(
n
)
:
If
P
(
n
)
exactly for which
n P
(
n
)
is true. For example:
n
= 1
;
2
;
3 =
)
P
(
n
)
.
1
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View Full DocumentCHAPTER 3: INDUCTION
Used to prove statements involving all or most of the natural numbers.
Whenever there are too many natural numbers to verify the statement
casebycase.
Principle of Induction:
Let
P
(
n
)
be a mathematical statement de
pending on the natural number
n
. Want to show that
P
(
n
)
is true for
all
n
2
N
(or for all
n
±
n
0
).
a) verify
P
(1)
is true (or
P
(
n
0
)
is true)
(basis step)
b) show that
P
(
n
) =
)
P
(
n
+ 1)
for (arbitrary)
n
2
N
(or for
n
±
n
0
)
(inductive step)
; here
P
(
n
)
is the
induction hypothesis
.
Then, by the principle of induction, the statement holds true
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 Spring '07
 GHEORGHICIUC
 Math, Mathematical Induction, Natural number, Inverse function, Commutativity, 1 g

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