离散数学CONDITIONAL STATEMENTS

Let p and q be propositions. The conditional statement p -> q is the proposition "if p, then
q ." The conditional statement p -> q is false when p is true and q is false, and true otherwise.
In the conditional statement p -> q, p is called the hypothesis (or antecedent or premise)
and q is called the conclusion (or consequence).
The statement p -> q is called a conditional statement because p -> q asserts that q is true
on the condition that p holds. A conditional statement is also called an implication.
　　The truth table for the conditional statement p -> q is shown in Table 5.

TABLE 5 The Truth Table for
the Conditional Statement p ->q.
p       q　　　　p->q
T       T 　　　　T
T 　　F　　 　　F
F 　　T 　　　　T
F 　　F　　　　 T

　　Note that the statement p -> q is true when both p and q are true and when p is false (no matter what truth
value q has).
　　Because conditional statements play such an essential role in mathematical reasoning, a variety of terminology is used to express p -> q. You will encounter most if not all of the following ways to express this conditional statement:

1. "if p, then q"  
2. "p implies q"   
3.  "if p, q"    
4. "p only if q"  
5.  "p is sufficient for q"                      
6. "a sufficient condition for q is p"  
7. "q if p"   
8. "q whenever p"
9. "q when p"                                    
10. "q is necessary for p"
11. "a necessary condition for p is q"     
12. "q follows from p"
13. "q unless -'p"

　　A useful way to understand the truth value of a conditional statement is to think of an obligation or a contract. For example, the pledge many politicians make when running for office is "If I am elected, then I will lower taxes."
　　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 mayor 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.
　　Many people find it confusing that "p only if q" expresses the same thing as "if p then q." To remember this, 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. A common error is for people to think that "q only if p" is a way of expressing p ->q. However, these statements have different truth values when p and q have different truth values.
　　The word "unless" is often used to express conditional statements. Observe that "q unless -'p" means that if -'p is false, then q must be true. That is, the statement "q unless -'p" is false when p is true and q is false, but it is true otherwise. Consequently, "q unless -'p" and p ->q
always have the same truth value.
　　We illustrate the translation between conditional statements and English statements in Example 7.

Example7：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 today, 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 today, but we do not go to the beach.

　　On the other hand, the statement

　　"If today is Friday, then 2 + 3 = 5."

　　is true from the definition of a conditional statement, because its conclusion is true. (The truth value of the hypothesis does not matter then.)

　　The conditional statement
　　"If today is Friday, then 2 + 3 = 6." is true every day except Friday, even though 2 + 3 = 6 is false.

　　We would not use these last two conditional statements in natural language (except perhaps in sarcasm), because there is no relationship between the hypothesis and the conclusion。