CS 381 Test



October, 2001



1. Fill in the blanks with the SHORTEST string of characters so that the resultant proposition is valid. [20]



$\neg (P \vee ( \neg P \wedge Q))$ $\Leftrightarrow$ ( $\framebox [1.0in]{$\neg P$}\wedge
\neg($ $ \framebox [1.0in]{$\neg P$}\wedge Q ))$


$\Leftrightarrow
($ $ \framebox [1.0in]{$\neg P$}\wedge ($ $\framebox [1.0in]{$P$}\vee \neg Q ))$


$\Leftrightarrow (($ $\framebox [1.0in]{$\neg P$}\wedge P) \vee
($ $ \framebox [1.0in]{$\neg P$}\wedge \neg Q ))$


$\Leftrightarrow
($ $\framebox [1.0in]{$F$}\vee
($ $ \framebox [1.0in]{$\neg P$}\wedge \neg Q ))$


$\Leftrightarrow ( \neg P \wedge \framebox [1.0in]{$\neg Q$}$ $ )$


$\Leftrightarrow \neg ( P$ $ \framebox [1.0in]{$\vee$}$ $ Q )$



2. Convert the following sentences to if_then sentences. No credit will be given if it is not in if_then form. [12]



(a) Whenever it is sunny, people are happy.
If it is sunny, then people are happy.
(b) People are happy only if it is sunny.
If people are happy, then it is sunny.
(c) It is necessary for people to be happy that it is sunny.
If people are happy, then it is sunny.



3. State each of the following formulas in English, if it is a wff. If it is not a wff, then give a reason why it is not a wff. Here $G(x,y)$ means $x$ is larger than $y$ and $N(x,y)$ means $x \neq y$ and the universe is the set of numbers: [15]



(a) $\forall x \exists y G(x,y)$
For every number x there is a number y such that x is larger than y. (Every number is larger than some number).
(b) $\exists x G(x,y) N(x,y)$
Not wff because there is no connective between G(x,y) and N(x,y).
(c) $\exists x \forall y [N(x,y) \rightarrow G(x,y)]$
There is a number x such that for every number y if x $\neq$ y,
then x $>$ y.
(There is a number which is greater than every other number).
(d) $\exists x \exists y N(x, G(x, y))$
Not wff because G(x,y) can not be an argument of N.
(e) $\exists x \forall y N(x,y) \rightarrow \exists x \exists y N(x,y)$
If there is a number which is not equal to any number then there is a number which is not equal to some number.



4 (a) Express the argument given below using the symbol suggested for each proposition. [6]
(b) Check whether or not the reasoning is correct using inference rules on the wffs (symbolic form) of (a). [12]



Argument: If I like mathematics (L), then I will study (S). Either I don't study or I don't pass mathematics ($\neg$P). If I don't pass mathematics, then I don't graduate ($\neg$G). But I graduate. Therefore, I like mathematics.



(a)
$L \rightarrow S$
$\neg S \vee \neg P$
$\neg P \rightarrow \neg G$
$G$
---------------
$L$



(b)
$\neg P \rightarrow \neg G$ third premise
$G$ fourth premise
------------------
$P$ by modus tollens


$P$
$\neg S \vee \neg P$ second premise
------------------
$\neg S$ by disjunctive syllogism


$\neg S$
$L \rightarrow S$ first premise
------------------
$\neg L$ by modus tollens


This contradicts the conclusion of the argument $L$. Thus the argument is not valid.



5. Express the assertions given below as a proposition of a predicate logic using the following predicates. The universe is the set of objects.[16]



$S(x, y)$: $x$ is sent along $y$.
$M(x)$: $x$ is a message.
$R(x)$: $x$ is a message route.
$ N(x) $: $x$ is non-faulty.



(a) Some message routes are faulty.
$\exists x [R(x) \wedge \neg N(x)]$
(b) Every message is sent along some message route.
$\forall x [M(x) \rightarrow \exists y [R(y) \wedge S(x,y)]]$
(c) Every message is sent along some message route if the route is non-faulty.
$\forall x [M(x) \rightarrow \exists y [R(y) \wedge [N(y) \rightarrow S(x,y)]]]$
(d) Every message is sent along only non-faulty message routes.
$\forall x \forall y [ [M(x) \wedge R(y) \wedge S(x,y)] \rightarrow N(y)]$



6. Find the power set of each of the following sets: [4]



(a) {$\emptyset$}
{$\emptyset$ , {$\emptyset$} }
(b) {{$1$} , {$\emptyset$}}
{ $\emptyset$ , {{$1$}}, {{$\emptyset$}} , {{$1$} , {$\emptyset$}} }



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



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