Unformatted text preview: Negating an Implication
We have seen that
~(pVq) is logically equivalent to ~pΛ~q .
And we have seen that
~(p Λq) is logically equivalent to ~p V ~q .
Question: How do you negate p => q? You might think that the negation of an
implication is another implication, but it isn’t!
In fact, since “p =>q” is logically equivalent to
the OR statement “~p V q” , the negation
“~(p=>q)” is logically equivalent to the negation
“~( ~p V q)” . This is the same as “~~p Λ ~q” .
The double negation ~(~p) is just p. Thus
“~(p=>q)” is logically equivalent to “p Λ ~q”. We have just learned that the negation of an
implication is an AND statement.
Example: Negate the statement “If it will rain I
will go to the movies.”
Here p: It will rain and q: I will go to the movies.
The negation of “p=>q” is “p Λ ~q”, which is:
“It will rain and I will not go to the movies.” Example: Negate the sentence: Studying daily
implies you will get good grades.
Let p: (You) will study daily
q: You will get good grades.
The given sentence is p=>q.
Its negation is p Λ ~q , which is the sentence:
“You will study daily and you will not get good
grades.” Converse and Contrapositive
In this lesson we learn two ways to create
a new implication from a given one.
Given a statement “If p then q” the
converse is the statement “If q then p”
obtained by switching the positions of p
Example: Let p and q be the statements
p: I have brown hair
q: I have brown eyes.
q: The implication “If p then q” is the statement
A: “If I have brown hair then I have brown eyes.”
The converse is the statement
B: “If I have brown eyes then I have brown hair.”
We will see that the truth or falsity of A has little
connection to the truth of falsity of B.
connection Here are the truth tables for A and for B:
F if p then q
T if q then p
T The last two columns are different, so the truth
table for A is not the same as that for B.
table It happens very frequently in ordinary speech
well as in mathematics that people confuse an
implication with its converse. Try to be aware of
this common but serious error so you don’t
The contrapositive of the implication
A: “If p then q”
is the statement
C: “If ~q then ~p”
C: To obtain the contrapositive of A, negate both p
and q and then reverse their positions.
Let p, q be the statements from before:
p: I have brown hair.
q: I have brown eyes.
A: If I have brown hair then I have brown eyes.
The contrapositive of A is:
B: If I don’t have brown eyes, then I don’t have
B: The contrapositive sounds complicated. Its
importance comes from the fact that the
contrapositive is logically equivalent to the
original implication! One is true precisely when
the other is true. Here are the truth tables.
q ~q ~p
T We see that the last two columns are identical,
which says that “p0q” and its contrapositive
“~q0~p” have the same truth values. In other
words, they are logically equivalent, although
they sound quite different.
Observe: the converse of the converse is the
original implication. And the contrapositive of
the contrapositive is the original implication.
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- Spring '10
- Logic, Contrapositive, Logical connective, Logical implication