CS 281 Test II

March 25, 1999

1. Find the powerset of each of the following sets: [10]

(a) $\emptyset$
(b) {$\emptyset$ , {$\emptyset$ }}

2. Find the following Cartesian products: [10]

(a) { $1,2 \} \times \{ 2,1 \} \times \{ \emptyset \}$
(b) $\emptyset \times \{\{ \emptyset \}, \emptyset \}$

3. Find the smallest four elements of the set S defined recursively as follows: [8]

Basis Clause: $0 \in S$
Inductive Clause: If $x \in S$, then $2x + 1 \in S$.
Extremal clause: Nothing is in S unless it is obtained from the above two clauses.

4. Recursively define each of the following sets:

(a) The set of positive integers congruent to 1 modulo 3. [10]
(b) { $2^{n} - 1 \mid n$ is a natural number.} [10]

5. Prove $( A - B) - C \subseteq A - C$ for sets A, B, and C. [15]

6. Indicate which of the following are true and which are false. [16]

(a) $\{x \} \subseteq \{\{x \}\}$
(b) { $x\} \in \{ x\}$
(c) $\{ x \} \in \{\{x \}\}$
(d) $\emptyset \in \{ x \}$

7. Express the assertions given below as a proposition of a predicate logic using the following predicates. The universe is the set of objects.[21]

CH(x): x is a check.
CS(x): x is cashable.
V(x): x is valid.

(a) Some checks are not valid.
(b) All invalid checks are not cashable.
(c) For a check to be cashable it is necessary that it is valid.