1. Basic Definitions of Set Theory
Lecture 25
Section 5.1
Mon, Mar 5, 2007

2. Disjoint Sets Sets A and B are disjoint if A  B = .
A collection of sets A1, A2, …, An are mutually disjoint, or pairwise disjoint, if Ai  Aj =  for all i and j, with i  j.

3. Examples The following sets are mutually disjoint. {0}
{…, -3, 0, 3, 6, 9, …} = {3k | k  Z}
{…, -2, 1, 4, 7, 10, …} = {3k + 1 | k  Z}
{…, -1, 2, 5, 8, 11, …} = {3k + 2 | k  Z}

4. Partitions
A collection of sets {A1, A2, …, An} is a partition of a set A if
A1, A2, …, An are mutually disjoint, and
A1  A2  …  An = A.

5. Examples {{0}, {1, 2, 3, …}, {-1, -2, -3, …}} is a partition of Z.

6. Example
For each positive integer n  N, define f(n) to be the number of distinct prime divisors of n.
For example,
f(1) = 0.
f(2) = 1.
f(4) = 1.
f(6) = 2.

7. Example Define Ai = {n  N | f(n) = i}.
Then A0, A1, A2, … is a (infinite) partition of N.
Verify that
Ai  Aj =  for all i, j, with i j.
A0  A1  A2  … = N.

8. Power Sets
Let A be a set. The power set of A, denoted P(A), is the set of all subsets of A.
If A = {a, b, c}, then P(A) = {, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}}.
What is P()?
What is P(P())?
What is P(P({a}))?
If A contains n elements, how many elements are in P(A)?

9. A  B = {(a, b) | a  A and b  B}.
Cartesian Products
Let A and B be sets. Define the Cartesian product of A and B to be
A  B = {(a, b) | a  A and b  B}.
R  R = set of points in the plane.
R  R  R = set of points in space.
What is A  ?
How many elements are in
{1, 2}  {3, 4, 5}  {6, 7, 8}?

10. Representing Sets in Software
Given a universal set U of size n, there are 2n subsets of U.
Given an register of n bits, there are 2n possible values that can be stored.
This suggests a method of representing sets in memory.

11. Representing Sets in Software
For simplicity, we will assume that |U|  32.
Let U = {a0, a1, a2, …, an – 1}.
Using a 32-bit integer to represent a set S, let bit i represent the element ai.
If i = 0, then ai  S.
If i = 1, then ai  S.
For example, represents the set S = {a0, a2, a3, a4, a7}.

12. Example: Sets.cpp