1. Section 3.2
More set operations

2. More operations
Sometimes an operation forms a new set with a different type of objects than in the original sets.
Examples. A = {1, 2, 3}, B = {3, 5} are sets of numbers.
A × B = { (1,3), (1,5), (2,3), (2,5), (3,3), (3,5) } is a set of ordered pairs.
P(B) = { {3, 5}, {3}, {5}, { } } is a set of sets

3. Definitions
A × B = { (a,b) : a  A and b  B}. This is called the Cartesian product of A and B.
P(A) = { S : S  A }. This is called the power set of A.

4. Practice Let A = {2, 3, 4} and B = {3, 5}. Determine: A × B B × A
(A  B) × (A  B)
P(B)
P(A  B)

5. Practice
Let A = {a,b,c,d} and B = {b,c,e}. How many elements are in each of the following sets?
A × B
B × A
(A  B) × (A  B)
P(A)
P(B)
P(A  B)

6. Practice
List the elements which belong to the following set:

7. Practice
List five elements that belong to the set:

8. Practice
List five elements that belong to the set:

9. Definition
A partition of a set A is a set S of non- empty subsets of A satisfying the following:
If P1 and P2 are two different members of S, then P1  P2 = Ø. (That is, distinct parts are disjoint.)
The union of all of the members of S is the entire set A.

10. Examples and practice Let A = {1, 2, 3, 4, 5, 6}.
{{1, 2}, {4, 3}, {5}, {6}} is a partition of A with 4 parts.
{{1, 5, 3, 4, 2, 6}} is a partition of A with just 1 part.
{{1, 3}, {4, 2}, {3, 5, 6}} is not a partition of A. Why not?
{1, 2, {3, 4}, {5, 6}} is not a partition of A. Why not?
{{ }, {1, 4, 2}, {3, 5, 6}} is not a partition of A. Why not?

11. More practice
Which of the following are partitions of Z, the set of all integers? For each that is not, say why not.
{ {3n + 1 : n  Z}, {3n + 2 : n  Z}, {3n : n  Z} }
{ {2n + 1 : n  Z}, {3n + 2 : n  Z}, {6n : n  Z} }
{ {3n + 1 : n  Z}, {6n + 2 : n  Z}, {9n : n  Z} }
{ {4n + 1 : n  Z}, {4n + 3 : n  Z}, {2n : n  Z} }