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Proof by Contradiction
Math 235 Fall 2000
Proof by contradiction is perhaps the strangest method of proof, since you start by assuming
that what you want to prove is
false
, and then show that something ridiculous happens 
thus your original assumption must have been false! For example, suppose you wish to prove
a statement
S
is true. You start a proof by contradiction by supposing that the statement
S
is
false
. Then you make logical arguments and conclusions that follow from this supposition
that
S
is false, until you arrive at a
contradiction
(for example, two logically contradictory
statements, or a statement that contradicts one of the assumptions of the problem). Since
the assumption that
S
is false led to a contradiction, the statement “
S
is false” must itself
be false; in other words,
S
must be true!
One ﬁnal note: Proof by contradiction should not be used if you can think of a more
“constructive” proof,
i.e.
a proof that starts only from the assumptions of the problem and
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This note was uploaded on 01/31/2011 for the course COMP 101 taught by Professor Rajivsir during the Winter '10 term at National.
 Winter '10
 RajivSir

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