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```CmSc 175 Discrete Mathematics
Homework 03: Logic, Arguments
due date: 02/10/11
1. Using the equivalence laws to show that
~(~(P Q) V P ) V Q  Q
(15 pts)
2. Write the contrapositive, converse, inverse and negation of the following
expressions. Apply De Morgan’s laws if necessary:
(12 pts)
a. P  Q V R
Contrapositive:
Converse:
Inverse:
Negation:
b. P  R  Q
Contrapositive:
Converse:
Inverse:
Negation:
3. Assuming that the premises are true, determine whether the arguments are valid
or invalid. If valid, indicate the inference rule. If invalid, indicate the type of the
error:
(8 pts)
3.1. Premises:
a. If I go to the movies, I won't finish my homework
b. If I don't finish my homework, I won't do well on the exam tomorrow.
Conclusion: If I go to the movies, I won't do well on the exam tomorrow.
Valid Invalid
1
3.2. Premises:
a. If John is not playing, the team will not win.
b. If the team does not win, the trip will be postponed.
c. John is playing.
Conclusion:
The trip is not postponed
Valid Invalid
4. Show how we can prove the truth of U from the premises below. Indicate the
inference rule at each step in the proof.
(20 pts)
(1) P  ~Q
(2) R  ~S
(3) ~P  S  U
(4) ~R  Q
(5) S
For example, the first step in the proof would be:
(6) ~R, by (2), (5) and Modus Tollens. Now we can add (6) to the premises, and
then combine (6) and (4) and apply MP to conclude
(7) Q
The idea is to look for a pair of premises that matches an inference rule, and then
apply that rule to make a new conclusion. See all inference rules in the table in
Lesson 04.
5. For each of the sentences below:

Represent the following sentences as quantified logical expressions using
the predicates : student(x), soccer_player(x), and healthy(x). (We can
assume that the domain is ‘all people’ and we do not indicate it explicitly)

Find the negation of the logical form
Example:
Students that play soccer are healthy.
Expression: x, student (x)  soccer_player(x),  healthy(x).
Negation: x, student(x)  soccer_player(x)  ~ healthy(x).
(18 pts)
2
a. All healthy students play soccer
Expression:
Negation:
b. Some healthy soccer players are students
Expression:
Negation:
c. No soccer players are healthy students
Expression:
Negation:
6. Indicate whether the following arguments in predicate logic are valid or invalid
(underline the correct answer). If valid, indicate the inference rule. If invalid,
indicate the type of the error:
(12 pts)
6.1.
6.2.
6.3.
Therefore peter is a teacher
Valid
Invalid
Valid
Invalid
Valid
Invalid
No lions like cherries
Peter likes cherries
Therefore Peter is not a lion
Pilots are happy but not wealthy
Peter is wealthy
Therefore Peter is not a pilot
7. Prove that the sum of two odd numbers is even
(15 pts)
3
```
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