CK-12-Pre-Calculus Concepts

http://www.ck12.org Chapter 16. Logic and Set Theory

TABLE16.15:(continued)

F T T F F

F F T T T

Notice that the truth values of the inverse are not identical to the truth values of the original conditional. The absence
of rain does not guarantee a dry driveway. Some children could have had a water balloon party in the summer.Just
because a statement is true does not mean that its inverse will be true!
The converse of a conditional statement switches the order of the hypothesis and the conclusion.
Converse:Q→P= If the driveway is wet, then it is raining.

TABLE16.16:

P Q Q→P

T T T

T F T

F T F

F F T

Notice that the truth values of the converse are also not identical to the truth values of the original conditional.
If children playing with water balloons made the driveway wet then it isn’t necessarily raining. Just because a
statement is true does not mean that its converse will be true!
The contrapositive of a conditional statement switches the hypothesis with the conclusion and negates both parts.
Contrapositive:∼Q→∼P= If the driveway is not wet, then it is not raining.

TABLE16.17:

P Q ∼P ∼Q ∼Q→∼P

T T F F T

T F F T F

F T T F T

F F T T T

The contrapositive of a conditional statement is functionally equivalent to the original conditional. This is because
you can logically conclude that a dry driveway means no rain. This means that if a statement is a true then its
contrapositive will also be true.
Example A
Assume the statement “everyone with blonde hair is smart” is true. Use the contrapositive to write another statement
that is related and also true.
Solution:The statement “everyone with blonde hair is smart” can be rewritten as “if a person has blonde hair then
the person is smart”. The contrapositive is “if a person is not smart, then the person does not have blonde hair”. This
statement must be true if the original statement is true.
Example B
Write the inverse, converse and contrapositive of the following conditional statement.
If you buy our product, then you are attractive.
Solution:Note that advertisers regularly imply certain results about their products that may or may not be true. If
you listen carefully you will notice that ironclad conditional statements are always avoided so they are not technically