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

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.

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}

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.

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

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.

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.

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)?

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}?

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.

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, 10011101 represents the set S = {a0, a2, a3, a4, a7}.

Example: Sets.cpp