CS 281 Final Exam

June 25, 1998

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]

D(x): x is a day.
P(x): x is a person.
R(x): x is rainy.
S(x): x is sunny.
L(x,y ): x likes y.


(a) All days are sunny.
(b) Some days are not rainy.
(c) It is always a sunny day only if it is a rainy day.
(d) Some people like only rainy days.
(e) Some people like no rainy days.

2. Prove that if $A \cup B = A - B$, then $B = \emptyset$. [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 power set of { 1,2,3,4,5,6,7}.
$(A,B) \in R$ iff $A \cap B \neq \emptyset$.
(b) S is the set of real numbers.
$(x,y) \in R$ iff $\mid x \mid \leq \mid y \mid$.
(c) S is the power set of { 1,2,3,4,5,6,7}.
$(A,B) \in R$ iff $\mid A \mid = \mid B \mid$.
(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. The n-th term of a sequence a_n is given by a_n = 1 + (-1)ⁿ⁺¹, where n represents positvie integers.

(a) Write the first three terms of the sequence. [5]
(b) Recursively define the sequence { $a_{n} \mid n$ is a positive integer.} [10]

5. Prove by mathematical induction that $\Sigma_{i = 1}^{n} i(i+1) = n(n+1)(n+2)/3$. [15]

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

(a) Find R². [5]
(b) Show that R² is symmetric. [5]

7 (a) Prove that if a binary relation R is symmetric, then R² is symmetric. [7]
(b) Prove that if a binary relation R is symmetric, then Rⁿ is symmetric for all positive integers n. You may use RⁿR = RRⁿ if necessary. [10]

Note that Rⁿ is defined recursively for positive integers n as follows:
Basis Clause: R¹ = R.
Inductive Clause: Rⁿ⁺¹ = RⁿR.