
We know that the negation of a
true statement will be
false, and the
negation of a false statement
will be true. But what happens
when we try to
negate a
compound statement?

Negating
a Conjunction (and)
and a Disjunction (or): 
The negation
of a conjunction (or disjunction) could be as simple as
placing the word "not" in front of the entire sentence.
Conjunction:
"Snoopy wears goggles and scarves."
"It is not the case that Snoopy wears
goggles and scarves."
While by our negation we know that
Snoopy does not wear BOTH goggles and scarves, we cannot say for
sure that he does not wear ONE of these items. We can only
state that he does not wear goggles or he does not wear scarves.
()
Disjunction:
"I will paint the room blue or green."
"It is not
the case that I will paint the room blue or
green."
If I am not painting the room blue or
green, then I am not painting EITHER color. So it can be said
that "I am not painting the room blue" and "I am not painting the
room green".
DeMorgan's Laws:
(negating
AND and
OR)
(The statements shown are logically equivalent.)

Notice that the negation symbol is
distributed across the parentheses and the symbols
are changed from AND to
OR (or vice versa). 

Negating
a Conditional (if ...
then): 
Remember: When working with a
conditional, the statement is only FALSE when the hypothesis
("if") is TRUE and the conclusion ("then") is FALSE.
"If 9 + 3 = 12,
then 9 is a prime number." is a
FALSE statement.
"It is not the case that
if 9 + 3 = 12, then 9 is a prime number." is TRUE.
"9 + 3 = 12 and 9 is
not a prime number." is a TRUE
statement.
Negate a Conditional:
(negating
IF ... THEN)

Notice that the statement is
rewritten as a conjunction and only the second
condition is negated. 

Negating
a Biconditional (if and
only if): 
Remember: When working with a
biconditional, the statement is TRUE only when both
conditions have the same truth value.
"A triangle has only 3 sides if and
only if a square has only 4 sides."
... is logically equivalent to ...
"If it is a triangle then it has only 3 sides
and if it is a square then it has only 4 sides."
To negate a
biconditional, we will negate its logically equivalent statement by
using DeMorgan's Laws and Conditional Negation.
Negate a Biconditional:
(negating
IF AND ONLY IF)

ALL 
SOME 
Consider: "ALL students are opera
singers."
(Meaning that there are NO students who are not
opera
singers.)

Consider: "Some
rectangles are squares."
(Meaning that there exists at least one rectangle that
is a square.) 
Negation: "It is
not true that ALL students are opera singers."
"SOME students are not opera
singers." 
Negation: "It
is not true that some
rectangles are squares."
"NO rectangles are squares." 
Negate ALL and SOME:
ALL A
are B.
negates to
SOME A are
not B.
SOME A
are B.
negates to
NO A are B. 
