### Homework #1

Some problems for this homework are from the textbook. Please note that I have changed the content of some of them.

1. For each collection of sets, find the smallest set A such that the collection is a subset of power(A).

a) {{{a}, {b}}, {c}} (5 points)
b) {{a, E}, {E}} where E is the empty set (5 points)
c) {E, {E}} (5 points)
d) {{a}, {{a}}} (5 points)

2. Assume that A and B are sets defined as

A = { x | x=4k-1 and k natural number }

B = { x | x=3k-5 and k natural number }

a) List the first 10 elements of the union of A and B (5 points)
b) List the first ten elements of the intersection of A and B (5 points)
c) List the first ten elements of A-B (5 points)
d) List the first ten elements of the symmetric difference between A and B (5 points)
e) Decide whether A is subset of B or not. (5 points)

3. 17 (page 28). Instead of the 'less than' sign (<) use the 'less than or equal' in the set definitions. (45 points)

4. 18 (page 28). Define the set An as follows

An = { x | x is natural number and x < n }

(25 points)

5. 23 (page 29). Assume that the cardinality of the union of the three sets is 281 instead of 280 and that the size (cardinality) of A is 99. (10 points)

6. 26 (page 29). Assume that there are 26 men and 20 noncity females. (20 points)

7. Find the union and the intersection of the following pairs of bags (30 points)

a) [x, x, y] and [x, y, z] (5 points)
b) [x, y] and [y, x, x, y] (5 points)
c) [a, a, [x, y]] and [x, y, [a, a]] (5 points)
d) [1, 2, 2, [a, [b]]] and [2, 1, 1, [b], [a], [[b]]] (5 points)

8. Find a bag A that solves the following two simultaneous bag equations:

A union [3, 2, 3, 2, 4] = [2, 2, 3, 3, 4, 4]
A intersect [2, 3, 2] = [2, 3]

(10 points)

9. 2 (page 55). Use A={x, y, z} and B={0, 1}. (30 points)

10. 7 (page 55) (25 points).

Maximum mark: 260 (100%)