CS 381 Test 1

October 7, 2003

1. Convert the following statements to if_then form in English:

(a) This store is closed on Wednesdays.
If it is Wednesday, the store is closed.
(b) Free discussions are necessary for further scientific advances.
If there are to be further scientific advances, there must be free discussions.
(c) You can have a healthy body only if your diet is healthy.
If you have a healthy body, then your diet must be healthy.
(d) Many people can not solve difficult problems. [20]
If problems are difficult, many people can not solve them.

2. Negate the following statements in English. Give a form other than simply putting "not" or "it is not the case that" in front:

(a) If today is Tuesday or Thursday, then I have a class at 3:00p.m.
Today is Tuesday or Thursday and I don't have a class at 3:00p.m.
(b) Everyone read some book on this subject.
Someone have not read any book on this subject.
(c) Either someone eats it or no one likes it.
Neither someone eats it nor no one likes it.
OR
No one eats it but someone likes it.
(d) It is awfull but he likes it. [20]
It is not awful or he does not like it.

3. Find the converse and contrapositive of the following statements in English:

(a) This program runs only if I am happy.
Converse: If I am happy then this program runs.
Contrapositive: If I am not happy, then this program does not run.
(b) I can't complete the work unless I get more help. [20] Converse: If I can't complete the work, then I don't get more help.
Contrapositive: If I can complete the work, then I get more help.

4. Find the dual of $[False \vee ( P \wedge \neg Q ) \vee \neg ( Q \wedge True)]$. [5]
$[ True \wedge (P \vee \neg Q ) \wedge \neg ( Q \vee False) ]$

5 (a) Express the argument given below using the symbold suggested for each proposition. [8]
(b) Check whether or not the reasoning is correct using inference rules on the propositions in symbolic form. [12]

Argument:
If I am healthy(H), then I play basketball(B). Either I don't play basketball or I am not happy. Also either I am full of energy (E) or I am happy (P). If I am full of energy, then I am happy. Therfore I am not healthy.

(a)
(1) $H \rightarrow B$
(2) $\neg B \vee \neg P$
(3) $E \vee P$
(4) $E \rightarrow P$
-------------
$\neg H$

(b)
(4) can be written as (5) $\neg E \vee P$.
Thus from (3) $\wedge$ (5) we get
$(E \vee P ) \wedge (\neg E \vee P)$
$\Leftrightarrow$ $(E \wedge \neg E ) \vee P$
$\Leftrightarrow$ $False \vee P$
$\Leftrightarrow$ $P$ -- (6)
From (2) and (6) by Disjunctive Syllogism, $\neg B$ follows.
Then from $\neg B$ and (1) by Modus Tollens, $\neg H$ is obtained. Hence this argument is correct.

6. Fill in the blanks with the shotest string: [15]

(a) $\neg [\hspace*{0.2cm} P \wedge Q \hspace*{0.2cm} ] \Leftrightarrow [ \hspace*... ...ace*{0.2cm} \framebox[1.0in] {$\vee$}\hspace*{0.2cm} \neg Q \hspace*{0.2cm} ]$

(b) $[ \hspace*{0.2cm} [ \hspace*{0.2cm} \neg P \wedge Q \hspace*{0.2cm}] \wedge [ ... ...} P \vee R \hspace*{0.2cm} ] \hspace*{0.2cm}] \hspace*{0.2cm} ] \rightarrow R$
$\Rightarrow [ \hspace*{0.2cm} \framebox[1.0in] {$\neg P$}\wedge [ \hspace*{0.2cm} P \vee R \hspace*{0.2cm} ] \hspace*{0.2cm} ] \rightarrow R$
$\Rightarrow [ \hspace*{0.2cm} \framebox[1.0in] {R}\hspace*{0.2cm} ] \rightarrow R$
$\Rightarrow \hspace*{0.2cm} \framebox[1.0in] {T}\hspace*{0.2cm}$

(c) $\neg P \rightarrow [ \hspace*{0.2cm} P \rightarrow Q \hspace*{0.2cm} ] \Leftr... ...m} \framebox[1.0in] {$\wedge$}\hspace*{0.2cm} P \hspace*{0.2cm} ] \rightarrow Q$
$\Leftrightarrow \hspace*{0.2cm} \framebox[1.0in] {False}\hspace*{0.2cm} \rightarrow Q$
$\Leftrightarrow \hspace*{0.2cm} \framebox[1.0in] {True}\hspace*{0.2cm}$

(d) $[ \hspace*{0.2cm} [ \hspace*{0.2cm} \neg P \rightarrow \neg Q \hspace*{0.2cm}]... ...hspace*{0.2cm}] \rightarrow [\hspace*{0.2cm} Q \rightarrow R \hspace*{0.2cm} ]$
$\Leftrightarrow [ \hspace*{0.2cm} [ \hspace*{0.2cm} Q \rightarrow \hspace*{0.2... ...pace*{0.2cm} ] \rightarrow [\hspace*{0.2cm} Q \rightarrow R \hspace*{0.2cm} ]$
$\Leftrightarrow [ \hspace*{0.2cm} Q \rightarrow \hspace*{0.2cm} \framebox[1.0i... ...hspace*{0.2cm}] \rightarrow [\hspace*{0.2cm} Q \rightarrow R \hspace*{0.2cm} ]$
$\Leftrightarrow \hspace*{0.2cm} \framebox[1.0in] {True}\hspace*{0.2cm}$