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Math 117 – March 31, 2011
2
Proof
Inductive reasoning
is the type of reasoning in which one draws a conclusion about the general
case from considering particular examples.
Deductive reasoning
is the type of reasoning in which one draws a conclusion by applying a general
principle to a particular case.
Inductive reasoning by itself does not constitute a proof. One needs to use a deductive argument to
prove the conclusion, even if the conclusion was ﬁrst obtained by inductive reasoning.
Most theorems can be formulated in the form
p
⇒
q
, in which case
p
is called the
hypothesis
and
q
is
called the
conclusion
. When constructing a proof of the implication
p
⇒
q
, one usually builds a logical
bridge of simpler implications:
p
⇒
p
1
⇒
p
2
··· ⇒ ··· ⇒
q
2
⇒
q
1
⇒
q.
The
contrapositive
of
p
⇒
q
is
∼
q
⇒∼
p
, and it is equivalent to the original implication, i.e.
(
p
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 Spring '08
 Akhmedov,A
 Math, Indirect Proof

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