CS 381 Final Exam




Fall 2001




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. [15]



$B(x)$: $x$ is a book.
$E(x)$: $x$ is expensive.
$W(x)$: $x$ is well-written.



(a) There are expensive books.
(b) A book is expensive only if it is well-written.
(c) Only well-written books are expensive.



2 (a) Express the argument given below in the symbolic form of first order predicate logic. Use the predicate symbols shown. You may also use the predicate = if needed. [5]
(b) Prove or disprove that the argument is valid by deriving the conclusion from the premises using inference rules. Use one rule for each step of your reasoning and state what rule is used. [10]



Argument: Every computer science student writes better computer programs than someone else. Everyone who earns less than anyone else does not write better computer programs than that person. Mary is a computer science student. Therefore Mary earns at least as much as someone else.



$C(x)$: $x$ is a computer science student.
$W(x,y)$: $x$ writes better computer programs than $y$.
$E(x,y)$: $x$ earns less than $y$.



3. Which of the following statements are true and which are false ? $A, B, C$ and $D$ are sets. [15]



(a) $(A - B) \cup (B - C) = ( A \cup B ) - C$
(b) If $A \cap B = A$, then $A = B$.
(c) If $A \subseteq B$ and $C \subseteq D$, then $A \cap C \subseteq B \cup C$.
(d) $A \subseteq (A - B) \cup (A - C)$
(e) $(A - B) \cup (B - A) = A \cup B$ if and only if $A = \emptyset$.



4. Prove by mathematical induction the following:



(a) $\Sigma_{i=1}^{n} ( 6 i^{2} + 5 )= 2n^{3} + 3n^{2} + 6n$ is an even number for all natural number $n$. [7]
(b) Prove that any amount of postage greater than or equal to 24 cents can be obtained using 5-cent and 7-cent stamps. [8]



5 (a) Recursively define the relation $\leq$ on the set of natural numbers. [7]
(b) Analogously to (a) recursively define the relation $\subseteq$ on the power set of the set of natural numbers. [8]



6. Fill in the table below with "Y" if the relation has the corresponding property, else with "N". In the table the following abbreviations are used.
Ref: Reflexive, Irref: Irreflexive, Antisym: Antisymmetric, Sym: Symmetric, Tran: Transitive. [15]



Relation Ref Irref Antisym Sym Tran
$\neq$ on sets          
$\leq$ on naturals          
$ <x, y> \in R$ iff $x$ was NOT born          
on the same day as $y$.          



7. What ordered pairs must be added to get the transitive closure of the relation R of the Question 6 above ? [5]



8. The $\Theta$ relation on the set of polynomials in $x$ is an equivalence relation. Find the equivalence class of $x^{2} - x + 2$. [5]




S. Toida 2004-04-27