CS 381 Final Exam

December 11, 2000

1. Using the predicate symbols given below, transcribe each of the following English sentences into a proposition of a predicate logic. The universe is the whole world. [20]

$B(x)$: $x$ is a book.
$D(x)$: $x$ is diligent.
$E(x)$: $x$ is expensive.
$S(x)$: $x$ is a student.
$L(x,y)$: $x$ likes $y$.

(a) Not everything is a book.
(b) All students are diligent only if they like books.
(c) For a student to like a book it is necessary that the book is not expensive.
(d) Some student likes only expensive books.
(e) No student likes expensive books.

2 (a) Translate into predicate logic wffs the argument given below using the given atomic formulas. Explain what inference rule is used in each step of your reasoning. [10]

Argument:
Every PC has a disk. Some PCs have a rewritable CD.
Therefore some PCs have both a disk and a rewritable CD.

Atomic Formulas to be Used:
$D(x)$: $x$ has a disk.
$P(x)$: $x$ is a PC.
$R(x)$: $x$ has a rewritable CD.

(b) Prove that the argument is correct by deriving the conclusion from the hypotheses step by step using one inference rule at a time. [10]

3. Prove by induction that $\Sigma_{i=1}^{n} i(i + 1) = n(n+1)(n + 2)/3$ for all natural number $n \geq 1$. [20]

4. Recursively define the set of natural numbers { $n^{2} - 1$ : $n \in N \wedge n \geq 1$}. [20]

5. Let $\cal F$ be a set of functions that map the set of real numbers to the set of real numbers. A binary relation $R_{BO}$ on $\cal F$ is defined as follows:

for any functions $f$ and $g$ in $\cal F$, $<f, g>$ $\in R_{BO}$
if and only if $f = \Theta (g)$.

Prove that $R_{BO}$ is an equivalence relation. [20]

Hint:
If $\lim_{x \rightarrow \infty} h_{1} (x)$ and $\lim_{x \rightarrow \infty} h_{2} (x)$ are finite, and not equal to $0$,
then $\lim_{x \rightarrow \infty} h_{1} (x) \ast h_{2} (x)$ = $\lim_{x \rightarrow \infty} h_{1} (x)$ $\ast$ $\lim_{x \rightarrow \infty} h_{2} (x)$,
and if $\lim_{x \rightarrow \infty} h_{1} (x) \neq 0$, then $\lim_{x \rightarrow \infty} 1/h_{1} (x)$ = $1/ \lim_{x \rightarrow \infty} h_{1} (x)$