LECTURE 10: COUNTEREXAMPLES AND CONTRADICTIONS (
§
5.15.2)
Great spirits have always encountered violent opposition from mediocre minds. Albert Einstein (1879
 1955)
1.
Counterexamples
We have often been dealing with proving universal statements like
R
:
∀
x
∈
S, P
(
x
)
.
But what if
R
is false? How do we prove that? If
R
is false, then its negation is true. So we want to
prove:
∼
R
:
∃
x
∈
S,
∼
P
(
x
)
.
We do this by ﬁnding some element
x
∈
S
where
P
(
x
) is false. This element
x
is called a
counter
example
. There may be more than one counterexample.
Result.
Show the following statement is false: For all
x
∈
R
,
x
2

x
6
= 0
.
Proof.
If
x
= 0 then
x
2

x
= 0. Thus
x
= 0 is a counterexample. (So is
x
= 1.)
±
We could ﬁx the previous statement if we use a diﬀerent domain: For all
x
∈
R
{
0
,
1
}
,
x
2

x
6
= 0.
Result.
Show the following statement is false: If
n
is an even integer then
n
2

n
is odd.
What is the negation of this statement? It is: There exists some
n
∈
Z
with
n
an even integer and
n
2

n
is even.
Proof.
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 Spring '11
 Smith
 Logic, Elementary arithmetic, Universal quantification

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