CS 281 Test II

June 12, 1998

1. Find the powerset of each of the following sets: [15]

(a) {1,2}
(b) { , { }}
(c) { { 1 }, { { 1 }}, 1 }

2. Find the following Cartesian products: [15]

(a) {
(b) {
(c)

3. List the members of each of the following sets: [10]

(a) { is even and }
(b) { is an integer satisfying }

4. Find the smallest four elements of the set S defined recursively as follows: [10]

Basis Clause:
Inductive Clause: If , then .
Extremal clause: Nothing is in S unless it is obtained from the above two clauses.

5. Recursively define each of the following sets:

(a) The set of non-negative integers divisible by 5. [7]
(b) { is a natural number.} [8]

6. Prove the following equalities on sets A, B, and C.

(a) [10]
(b) [15]

7. Indicate which of the following are true and which are false. [10]

(a) {
(b)
(c)
(d)

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