CS 381 Test I

October 4, 2000

1. Fill in the blanks with the SHORTEST string of characters so that the resultant proposition is valid. [35]

(a) $(P \wedge Q) \rightarrow R \Leftrightarrow$ $\neg (P \wedge \framebox [1.0in]{?}$ ) $\vee R$
$\Leftrightarrow$ ( $\neg P \vee \framebox [1.0in]{?}$ ) $\vee R$
$\Leftrightarrow$ ($\neg P \vee R$) $\vee \framebox [1.0in]{?}$
$\Leftrightarrow$ ( $P \wedge \neg R$) $\rightarrow \framebox [1.0in]{?}$
(b) $( (P \wedge (P \vee Q)) \Leftrightarrow$ $(P \vee \framebox [1.0in]{?}$ ) $\wedge ( P \vee Q )$
$\Leftrightarrow P \vee$ ( $\framebox [1.0in]{?}$ $\wedge Q$) $\Leftrightarrow P \vee$ $\framebox [1.0in]{?}$ $\Leftrightarrow P$

2. State the following formula in English, where $P(x,y)$ means $x \geq y$ and $Q(x,y)$ means $xy \geq 0$ and the universe is the set of numbers: [15]

(a) $\forall x \forall y P(x,y)$
(b) $\exists x \forall y [P(x,y) \rightarrow Q(x,y)]$
(c) $\forall x \exists y P(x,y) \wedge \exists x \exists y Q(x,y)$

3. Which ones of the following strings are wffs of the predicate logic ? Answer with 'yes' or 'no'. [21]

(a) $\forall x \exists y Q(x,y)$
(b) $\exists y \forall Q(x,y)$
(c) $P(\forall x) \wedge \exists x \forall y Q(x,y)$
(d) $\forall x P(x) \rightarrow \exists y Q(x,y)$
(e) $Q(P(x), y) \vee P(x)$
(f) $\neg P(3)$
(g) $P(x) \vee \forall x \exists y Q(x,y)$

4 (a) Express the argument given below using the symbol suggested for each proposition. [12]
(b) Check whether or not the reasoning is correct. Give your reasons. [17]

Russia was a superior power, and either France was not strong or Napoleon made an error. Napoleon did not make an error, but if the army did not fail, then France was strong. Hence the army failed and Russia was a superior power.

$R$: Russia was a superior power.
$F$: France was strong.
$N$: Napoleon made an error.
$A$: The army failed. --->