CS 281 Test I

February 16, 1999

1 (a) $\exists x [E(x) \wedge I(x)]$
(b) $\neg \forall x [D(x) \wedge E(x)]$
(c) $\neg D(Rosen)$
(d) $\forall x \neg E(x)$ or equivalently $\neg \exists x E(x)$.

2 (a) $\vee$
(b) $\neg Q$, T
(c) P
(d) $\wedge$

3. If __ then __ form:
(a) If the diet is healthy, then the body is healthy.
(b) If the technology advances further, there are (must be) free discussions.
(c) If fuel can be saved, then there is good insulation or storm window throughout

Contrapositive:
(a) If the body is not healthy, then the diet is not healthy.
(b) If there are no free discussions, then the technology does not advance any further.
(c) If there is no good insulation or storm window throughout, then no fuel can be saved.

4(a)
$S \rightarrow C$
$S \vee R$
$\neg C$
-----------
R

(b)
$S \rightarrow C$
$\neg C$
-----------
$\neg S$
$S \vee R$
-----------
R

(c)
$T \rightarrow C \vee B$
$B \rightarrow \neg T$
$S \rightarrow \neg C$
T
-----------
S

First
$B \rightarrow \neg T$
T
-----------
$\neg B$

Then
$T \rightarrow C \vee B$
T
-----------
$C \vee B$
$\neg B$
-----------
C
$S \rightarrow \neg C$
-----------
$\neg S$

Hence the given conclusion is not correct, that is the reasoning is incorrect.