1549901369-Elements_of_Real_Analysis__Denlinger_

A .l The Logic of Propositions 589

Thus, '""P has the opposite truth-value from P.

Examples A.1.13 Some negations:

(a) The negation of "This car can go 97 mph." is "This car can't go 97

mph."

(b) The negation of "He's taking English and math" is "He is not taking

both English and math (although he might be taking one of them)." A smoother

way of saying that is, "Either he is not taking English or he is not taking math."

We shall say more about this when we discuss de Morgan's law below. 0

EXERCISE SET A.1-A

PART A: In Exercises 1- 12, a compound proposition is given. Assign variables,
like P, Q, etc. to the constituent propositions and translate the given verbal
sentence into logical symbols using the logical connectives we have studied.
Example: I will come to your party only if I don't have to work and I'm
feeling better.
Solution: Let C denote "I will come to your party;" W denote "I have
to work;" and F denote "I'm feeling better." Then the given proposition is
symbolized:

C ~ ('"" W /\ F).

1. 9 is neither an even number nor a prime number.


2. I like peanuts but not walnuts. [Note: here, "but" is a conjunction used
   to mean "and."]


3. I like ham and beans but not for breakfast.


4. You will get an A only if you work hard.


5. Analysis is an interesting subject, and I will get an A if the professor is
   easy.


6. A triangle is isosceles if it is equilateral.


7. If you don't show up for the final exam, or if you fail it, you will not pass
   the course.


8. To receive a passing grade for this course you must pay your bill, come
   to class, do all the assignments, and pass the tests.


9. To be elected President, you must have strong party backing and lots of
   money, but cannot have a record of dishonesty or be seen as ignorant of
   the issues.