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PROOF TECHNIQUES
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An identity is a mathematical sentence that is always true.
For example, x+x=2x is an identity.
A contradiction is a mathematical sentence that is always false.
For example, x=1 and x&neq;1 is a contradiction.
Given any mathematical statement S ,
S or
(not S) is an identity, and
S and
(not S) is a contradiction,
as the truth table below shows:
| S |
not S |
S or (not S) |
S and (not S) |
| T | F | T | F |
| F | T | T | F |
The proof by contradiction technique can be applied to any type of mathematical sentence:
PROOF BY CONTRADICTION:
To prove that a sentence is true, you can assume that it is false, and then
reach a contradiction.
|
This approach is justified by the truth table below, where it is
shown that S is equivalent to
(not S) ⇒ (a contradiction) :
| S |
not S |
a contradiction |
(not S) ⇒ (a contradiction) |
| T | F | F | T |
| F | T | F | F |
There are three standard approaches for proving an implication,
A⇒B :
DIRECT PROOF:
To prove that an implication A⇒B
is true,
we can assume that A is true, and
show that B is also true.
|
The next method is based on the fact that an implication is equivalent to its contrapositive.
PROOF BY CONTRAPOSITION:
To prove that an implication A⇒B
is true,
we can alternately prove that its contrapositive
(not B)⇒
(not A)
is true:
Assume that B is false, and
show that A is also false.
|
Applying "proof by contradiction" to an implication leads to a proof technique that
is called an "indirect proof":
INDIRECT PROOF:
To prove that an implication A⇒B
is true,
we can assume that A is true, but B is false,
and then reach a contradiction.
|
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