Discrete Mathematics- An Open Introduction - 3rd Edition, 2016a

10 0. Introduction and Preliminaries
This sort of argument shows up outside of math as well. If you ever
found yourself starting an argument with “hypothetically, let’s assume
. . . ,” then you have attempted a direct proof of your desired conclusion.
An implication is a way of expressing a relationship between two statements.
It is often interesting to ask whether there are other relationships
between the statements. Here we introduce some common language to
address this question.
Converse and Contrapositive.
• The converse of an implication P → Q is the implication Q →
P. The converse is NOT logically equivalent to the original
implication. That is, whether the converse of an implication is
true is independent of the truth of the implication.
• The contrapositive of an implication P → Q is the statement
¬Q → ¬P. An implication and its contrapositive are logically
equivalent (they are either both true or both false).
Mathematics is overflowing with examples of true implications which
have a false converse. If a number greater than 2 is prime, then that
number is odd. However, just because a number is odd does not mean it
is prime. If a shape is a square, then it is a rectangle. But it is false that if a
shape is a rectangle, then it is a square.
However, sometimes the converse of a true statement is also true. For
example, the Pythagorean theorem has a true converse: if a 2 + b 2 c 2 ,
then the triangle with sides a, b, and c is a right triangle. Whenever you
encounter an implication in mathematics, it is always reasonable to ask
whether the converse is true.
The contrapositive, on the other hand, always has the same truth value
as its original implication. This can be very helpful in deciding whether
an implication is true: often it is easier to analyze the contrapositive.
Example 0.2.5
True or false: If you draw any nine playing cards from a regular
deck, then you will have at least three cards all of the same suit. Is
the converse true?
Solution. True. The original implication is a little hard to analyze
because there are so many different combinations of nine cards. But
consider the contrapositive: If you don’t have at least three cards all
of the same suit, then you don’t have nine cards. It is easy to see
why this is true: you can at most have two cards of each of the four
suits, for a total of eight cards (or fewer).