CS 281 Final Exam

August 6, 1999

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

C(x): x is a car.
E(x): x is expensive.
S(x): x is slow.
T(x): x is a train.
ST(x,y ): x is slower than y: y is faster than x.


(a) Some cars are slow but not expensive.
(b) Only cars are slow.
(c) A car is not expensive only if it is slow.
(d) Some trains are slower than any car.
(e) Some expensive cars are faster than some trains.

2 (a) Prove that $A \cup \emptyset = A$. [10]
(b) Is A equal to B if A - B = B - A ? Justify your answer. [10]

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 x² - y² = 0.
(b) S is a collection of sets.
$<A,B> \in R$ iff A - B = $\emptyset$.
(c) S is the set of real numbers.
$<x,y> \in R$ iff x*y is an even number.
(d) S is a collection of sets.
$<A,B> \in R$ iff $A \subseteq B$.
(e) S is a set of people.
$<x,y> \in R$ iff x and y take some courses together.

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

4. Prove the following by mathematical induction:

(a) 0² + 2² + 4² + ... + (2n)² = 2n( n + 1 )( 2n + 1 )/3. [10]
(b) n³ - n is an even number if $n \geq 0$. [10]

5. Let R be the relation on the set of propositions defined as follows:
$<p, q> \in R$ if and only if $p \Leftrightarrow q$. Answer the following questions:

(a) Prove that R is an equivalence relation. [9]
(b) Give two different examples of equivalence class of R. [6]
(c) What are the members of the equivalence class [True] ? [3]
(d) Let X be a symbolic logical form for a proposition in English (such as $p \rightarrow q$). Which of the following eight propositions are in the same equivalence class as X ? [12]
$X \wedge X$, $X \vee X$, $X \vee Y$ ( $X \not\Leftrightarrow Y$), $X \rightarrow X$,
$True \rightarrow X$, $X \vee [X \wedge \neg X ]$, $\neg X \rightarrow X$ and $\neg \neg X$.