CS 281 Final Exam



May 6, 1999



1. Express the assertions given below as a proposition of a predicate logic using the following predicates. The universe is the set of objects.[15]



I(x): x is interesting.
L(x): x is long.
M(x): x is a mystery.
S(x): x is a spy novel.
B(x,y ): x is better than y.


(a) Some spy novels are long but interesting.
(b) Only mysteries are interesting.
(c) A mystery is not long only if it is interesting.
(d) Some mysteries are better than all spy novels.
(e) Some long spy novels are better than any mysteries.



2. Prove that if $A \cap ( \overline{B} \cap \overline{C} ) = A$, then $B = C = \emptyset$. [13]



3. Test the following binary relation R on the given set S for reflexivity, irreflexivity, antisymmetry, symmetry and transitivity. Fill in the table below. [15]



(a) S is the set of real numbers.
$(x,y) \in R$ iff $\mid x \mid - \mid y \mid$ = 0.
(b) S is the power set of { 1,2,3,4,5,6,7}.
$(A,B) \in R$ iff $\mid A \mid \leq \mid B \mid$, where $\mid A \mid$ is the size of set A.
(c) S is the power set of { 1,2,3,4,5,6,7}.
$(A,B) \in R$ iff $A \cap B = \emptyset$.
(d) S is the set of natural numbers.
$(x,y) \in R$ iff x + y is even.
(e) S is the power set of { 1,2,3,4,5,6,7}.
$(A,B) \in R$ iff $A \subseteq B $.


Question Reflexive Irreflexive Antisymmetric Symmetric Transitive
(a)          
(b)          
(c)          
(d)          
(e)          

4. Let S be the set of strings consisting of an even number of a's and an arbitrary number of b's. Assume that the strings of S have at least one a or b.



(a) What is the shortest string of S ? [2]
(b) Give a shortest string that has at least one a and at least one b. How many such strings exist ? [5]
(c) Define S recursively. [10]



5. Prove by mathematical induction that $3^{n} \leq n$ ! for all integer $n \geq 7$. [15]



6. Let $R = \{(1,2), (2,1), (1,3), (3,1)$}.



(a) Find R3. [5]
(b) Show that R3 is symmetric. [5]



7. Prove that if a binary relation R is symmetric, then R3 is symmetric. [15]



Hints: (1) R2 is symmetric if R is symmetric. (2) R3 = R2R = RR2.


 

S. Toida
1999-06-24