LECTURE 12: PROVE OR DISPROVE (
§
7.3)
The reverse side also has a reverse side. Japanese Proverb
1.
Testing Statements
Sometimes statements are given for which we do not immediately know whether they are true or
false.
We can test them, and try to do a proof, and eventually we hope that we can gain enough
understanding to either prove or disprove the statement. There are many examples of this (in fact, this
is what mathematicians do!). There is the 4 color theorem, the conjecture on Fermat numbers, and the
odd perfect number problem.
Idea: Try a few examples, then try to sketch a proof. Make sure to cover some of the tricky cases in
your examples.
Example 1.
Prove or disprove: If
x
and
y
are real numbers,

x
+
y

=

x

+

y

.
Because we are dealing with absolute values, we try some negative numbers. If
x
= 1 and
y
=

1
then

x
+
y

=

0

= 0 and

x

= 1 and

y

= 1.
Proof.
The statement is false. Let
x
= 1 and
y
=

1. Then

x
+
y

= 0 but

x

+

y

= 2.
Example 2.
Prove or disprove: If
x, y, z
∈
Z
then two of these integers have the same parity.
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 Spring '11
 Smith
 Scientific evidence, Legal burden of proof

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