1
Negations
1.1
Welcome
Welcome to the eighth module in our course on proofs, “Negations.” Before going on, please do the
assigned readings.
1.2
Introduction
You will frequently encounter the negation of statement
A
. The
negation
of the statement, is the
statement
NOT
A
, not surprisingly. Because statements cannot be both true and false, exactly one
of
A
and
NOT
A
can be true.
1.3
Negating Statements
In some instances, finding the negation of a statement is easy. For example, if
A
is the statement
A
:
f
(
x
) has a real root.
its’ negation
NOT A
is the statement
NOT
A
:
f
(
x
) does not have a real root.
When
A
already contains a
NOT
we use the rule that two negatives is a positive, or that one
NOT
cancels another
NOT
. For example
A
:
7 is not a divisor of 28.
NOT
A
:
7 is a divisor of 28.
A specific rule applies when negating a statement containing the word
AND
. For example,
A
:
B
AND
C
NOT
A
:
NOT
(
B
AND
C
) which is equivalent to (
NOT
B
) OR (
NOT
C
)
Note that the connecting word has changed from
AND
to
OR
and that each term in the expression
has been negated.
The brackets are not needed because
NOT
precedes
OR
in logical evaluation,
but the brackets are useful to emphasize the change. Here is a specific example.
A
:
4
T
is isosceles and has perimeter 42.
NOT
A
:
4
T
is not isosceles or does not have perimeter 42.
Similar to the conjunctive
AND
, a specific rule applies when negating a statement containing
the word
OR
.
A
:
B
OR
C
NOT
A
:
NOT
(
B
OR
C
) which is equivalent to
NOT
B
AND
NOT
C
Note that the connecting word has changed from
OR
to
AND
and, again, each term in the
expression has been negated. Again, the brackets are not needed because
NOT
precedes
AND
in
logical evaluation, but the brackets are useful to emphasize the change. Here is a specific example
A
:
4
T
is isosceles or it has perimeter 42.
NOT
A
:
4
T
is not isosceles and it does not have perimeter 42.
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 Spring '08
 ANDREWCHILDS
 Logic, ObjectOriented Programming, Philosophy of mathematics, Quantification, Universal quantification

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