epdf.pub_the-nuts-and-bolts-of-proofs-third-edition-an-intr

20 The Nuts and Bolts of Proof, Third Edition
with and which is true or false exactly when the original statement is true or
false.
Given the statement A, we can construct the statement "not A," which is
false when A is true and true when A is false; "not A" is the negation of A.
Clearly, these two statements are related, but they are not logically
equivalent.
As most mathematical statements are in the form "If A, then B," we will
work in detail on the three statements related to "If A, then B," and defined
as follows:
• The converse of the statement "If A, then B" is the statement "If B,
then A." (To obtain the converse, reverse the roles of the hypothesis
and the conclusion.)
• The inverse of the statement "If A, then B" is the statement "If 'not A,'
then 'not B.'" (To obtain the inverse, negate the hypothesis and the
conclusion.)
• The contrapositive of the statement "If A, then B" is the statement "If
'not B,' then 'not A.'" (To obtain the contrapositive, construct the
converse, and then consider the inverse of the converse. This means
reverse the roles of the hypothesis and the conclusion and negate
them.)
Let us consider an example to clarify these definitions. Let the original
statement be:
If X is a rational number, then x^ is a rational number.
Its converse is the statement "If X2 is a rational number, then x is a
rational number."
Its inverse is the statement "If x is not a rational number, then X2 is not a
rational number."
Its contrapositive is the statement "If X2 is not a rational number, then x is
not a rational number."
These statements cannot be all logically equivalent because the original
statement and its contrapositive are true (prove that this claim is indeed
correct), while the converse and the inverse are false (why?). To avoid
guessing if and when these four statements are logically equivalent, we
will construct their truth tables. Because we want to compare the four
tables, the columns for the statements A and B must always be the
same. We want to know when, under the same conditions for A and B,
we get the same conclusions regarding the truth of the final composite
statements.