CS 395 Test

October 15, 1999

1. Express the assertions given below as a proposition of a predicate logic using the following predicates. The universe is the set of objects.[15]

F(x): x is a flower.
R(x): x is red.
P(x): x is a person.
L(x,y ): x likes y.


(a) Not everyone likes a flower.
(b) Everyone likes a flower if it is red.
(c) Some people like a flower only if it is red.
(d) Everyone likes some flowers.
(e) Some red flowers like only people.

2. State in English the negation of each of the propositions given below. Give a form other than simply putting "not" in front. [12]

(1) If it is raining, it is not snowing.
(2) Every student in this class has taken exactly two math courses at this school.
(3) It is not snowing only if it is raining.
(4) Someone has read every book in the library.

3. Find the converse and the contrapositive of the following statements. State them in English.[16]

(1) If it is raining, it is not snowing.
(2) It is not snowing only if it is raining.
(3) It can not be true that everything is expensive and everything breaks easily .
(4) It is necessary for taking this course that one has taken at least two mathematics courses.

4(a) Express the argument given below using the symbol indicated for each proposition. [3]
(b) Using the symbols of (a) for the propositions, explain how the reasoning proceeds i.e. identify each application of inference rule in the argument. What conclusion do you draw ? It may be an if-then sentence. Give your reasons. [10]

Argument:
If a book is unhealthy in tone ($\neg$H), then I do not recommend it for reading ($\neg$R); if a book is bound (B), then it is well-written (W); if a book is about travel (T), then it is healthy in tone; if a book is not bound, then I do not recommend you to read it.

5. Find the powerset of each of the following sets: [6]

(a) $\emptyset$
(b) {$\emptyset$}
(c) {$\emptyset$ , {$\emptyset$}}

6. Find the following Cartesian products: [8]

(a) { $1 , 2, 3 \} \times \emptyset$
(b) { $\emptyset \} \times \{ \emptyset \} \times \{ 1,2 \}$
(c) { $1, 2 \} \times \{ 3 \} \times \{ 1 \} \times \{ 1,2 \}$
(c) $\emptyset \times \{ \{ \emptyset \}, \emptyset \}$

7. Indicate which of the following are true and which are false. [15]

(a) { $x\} \subseteq \{ x\}$
(b) $\emptyset \in \{ x \}$
(c) $\emptyset \subseteq \{ x \}$
(d) $x \in \{\{x \}\}$
(e) $\{x \} \subseteq \{\{x \}\}$

8. Which rules of inference are used to establish the conclusion of the following argument ? [15]

Premises:
"No monkeys are soldiers."
"Some monkeys are mischievous."

Conclusion:
"Some mischievous creatures are not soldiers."

Give your answer using M(x), C(x) and S(x) to denote "x is a monkey", "x is mischievous" and "x is a soldier", respectively, and assuming the universe is the set of all creatures. Using these symbols, the statements given above can be expressed as follows:

$\forall x [M(x) \rightarrow \neg S(x) ]$
$\exists x [M(x) \wedge C(x) ]$
---------------------
$\exists x [C(x) \wedge \neg S(x) ]$

You may use the following table.

Logical Equivalences and Implications

1. $P \Leftrightarrow (P \vee P)$	2. $P \Leftrightarrow (P \wedge P)$
3. $(P \vee Q) \Leftrightarrow (Q \vee P)$	4. $(P \wedge Q) \Leftrightarrow (Q \wedge P)$
5. $[(P \vee Q) \vee R] \Leftrightarrow [P \vee (Q \vee R)]$	6. $[(P \wedge Q) \wedge R] \Leftrightarrow [P \wedge (Q \wedge R)]$
7. $\neg (P \vee Q) \Leftrightarrow (\neg P \wedge \neg Q)$	8. $\neg (P \wedge Q) \Leftrightarrow (\neg P \vee \neg Q)$
9. $[P \wedge (Q \vee R)] \Leftrightarrow [(P \wedge Q \vee (P \wedge R)]$	10. $[P \vee (Q \wedge R)] \Leftrightarrow [(P \vee Q) \wedge (P \vee R)]$
11. $(P \vee T) \Leftrightarrow T$	12. $(P \wedge T) \Leftrightarrow P$
13. $(P \vee F) \Leftrightarrow P$	14. $(P \wedge F) \Leftrightarrow F$
15. $(P \vee \neg P) \Leftrightarrow T$	16. $(P \wedge \neg P) \Leftrightarrow F$
17. $P \Leftrightarrow \neg (\neg P )$	18. $(P \rightarrow Q) \Leftrightarrow (\neg P \vee Q)$
19. $(P \leftrightarrow Q) \Leftrightarrow [(P \rightarrow Q) \wedge (Q \rightarrow P)]$	20. $[(P \wedge Q) \rightarrow R] \Leftrightarrow [P \rightarrow (Q \rightarrow R)]$
21. $[(P \rightarrow Q) \wedge (P \rightarrow \neg Q)] \Leftrightarrow \neg P$	22. $(P \rightarrow Q) \Leftrightarrow (\neg Q \rightarrow \neg P)$
1. $P \Rightarrow (P \vee Q)$	2. $(P \wedge Q) \Rightarrow P$
3. $[P \wedge (P \rightarrow Q)] \Rightarrow Q$	4. $[(P \rightarrow Q) \wedge \neg Q] \Rightarrow \neg P$
5. $[\neg P \wedge (P \vee Q)] \Rightarrow Q$	6. $[(P \rightarrow Q) \wedge (Q \rightarrow R)] \Rightarrow (P \rightarrow R)$
7. $(P \rightarrow Q) \Rightarrow [(Q \rightarrow R) \rightarrow (P \rightarrow R)]$	8. $[(P \rightarrow Q) \wedge (R \rightarrow S)] \Rightarrow [(P \wedge R) \rightarrow (Q \wedge S)]$
9. $[(P \leftrightarrow Q) \wedge (Q \leftrightarrow R)] \Rightarrow (P \leftrightarrow R)$	10. $[(P \rightarrow Q) \wedge (R \rightarrow S)] \Rightarrow [(P \vee R) \rightarrow (Q \vee S)]$)

Additional Inference Rule: Conjunction

P
Q
------------------
P $\wedge$ Q