Page 28 - Discrete Mathematics and Its Applications

1.1 Propositional Logic 7

If the politician is elected, voters would expect this politician to lower taxes. Furthermore, if the
politician is not elected, then voters will not have any expectation that this person will lower
taxes, although the person may have sufficient influence to cause those in power to lower taxes.
It is only when the politician is elected but does not lower taxes that voters can say that the
politician has broken the campaign pledge. This last scenario corresponds to the case when p
is true but q is false in p → q.
Similarly, consider a statement that a professor might make:
“If you get 100% on the final, then you will get an A.”
If you manage to get a 100% on the final, then you would expect to receive an A. If you do not
get 100% you may or may not receive an A depending on other factors. However, if you do get
100%, but the professor does not give you an A, you will feel cheated.
Of the various ways to express the conditional statement p → q, the two that seem to cause
the most confusion are “p only if q” and “q unless ¬p.” Consequently, we will provide some
guidance for clearing up this confusion.
To remember that “p only if q” expresses the same thing as “if p, then q,” note that “p only
if q” says that p cannot be true when q is not true. That is, the statement is false if p is true,
but q is false. When p is false, q may be either true or false, because the statement says nothing
about the truth value of q. Be careful not to use “q only if p” to express p → q because this is
incorrect. To see this, note that the true values of “q only if p” and p → q are different when
p and q have different truth values.
You might have trouble
To remember that “q unless ¬p” expresses the same conditional statement as “if p, then
understanding how
“unless” is used in q,” note that “q unless ¬p” means that if ¬p is false, then q must be true. That is, the statement
conditional statements “q unless ¬p” is false when p is true but q is false, but it is true otherwise. Consequently,
unless you read this “q unless ¬p” and p → q always have the same truth value.
paragraph carefully.
We illustrate the translation between conditional statements and English statements in Ex-
ample 7.
EXAMPLE 7 Let p be the statement “Maria learns discrete mathematics” and q the statement “Maria will
find a good job.” Express the statement p → q as a statement in English.

Solution: From the definition of conditional statements, we see that when p is the statement
“Maria learns discrete mathematics” and q is the statement “Maria will find a good job,” p → q
represents the statement

“If Maria learns discrete mathematics, then she will find a good job.”
There are many other ways to express this conditional statement in English. Among the most
natural of these are:

“Maria will find a good job when she learns discrete mathematics.”
“For Maria to get a good job, it is sufficient for her to learn discrete mathematics.”
and
“Maria will find a good job unless she does not learn discrete mathematics.”
▲

Note that the way we have defined conditional statements is more general than the meaning
attached to such statements in the English language. For instance, the conditional statement in
Example 7 and the statement
“If it is sunny, then we will go to the beach.”
are statements used in normal language where there is a relationship between the hypothesis
and the conclusion. Further, the first of these statements is true unless Maria learns discrete
mathematics, but she does not get a good job, and the second is true unless it is indeed sunny,
but we do not go to the beach. On the other hand, the statement