Basic Definitions of Set Theory Lecture 23 Section 5.1 Wed, Mar 8, 2006

The Universal Set Whenever we use sets, there must be a universal set U which contains all elements under consideration. Typical examples are U = R and U = N. Without a universal set, taking complements of set is problematic.

Set Operations Let A and B be set. Define the intersection of A and B to be A  B = {x  U | x  A and x  B}. Define the union of A and B to be A  B = {x  U | x  A or x  B}. Define the complement of A to be A c = {x  U | x  A}.

Set Operations Notice that the set operations of intersection, union, and complement correspond to the boolean operations of and, or, and not.

Set Differences Define the difference A minus B to be A – B = {x  U | x  A and x  B}. Define the symmetric difference of A and B to be A  B = (A – B)  (B – A).

Set Differences Do the operations of difference and symmetric difference correspond to boolean operations?

Subsets A is a subset of B, written A  B, if  x  A, x  B. A equals B, written A = B, if  x  A, x  B and  x  B, x  A. A is a proper subset of B, written A  B, if  x  A, x  B and  x  B, x  A.

Sets Defined by a Predicate Let P(x) be a predicate. Define a set A = {x  U | P(x)}. For any x  U, If P(x) is true, then x  A. If P(x) is false, then x  A. A is the truth set of P(x).

Sets Defined by a Predicate Two special cases. What predicate defines the universal set? What predicate defines the empty set?

Intersection and Union Let P(x) and Q(x) be predicates and define A = {x  U | P(x)}. B = {x  U | Q(x)}. Then the intersection of A and B is A  B = {x  U | P(x)  Q(x)}. The union of A and B is A  B = {x  U | P(x)  Q(x)}.

Complements and Differences The complement of A is A c = {x  U |  P(x)}. The difference A minus B is A – B = {x  U | P(x)   Q(x)}. The symmetric difference of A and B is A  B = {x  U | P(x)  Q(x)}.

Subsets A is a subset of B if  x  U, P(x)  Q(x), or  x  A, Q(x). A equals B if  x  U, P(x)  Q(x), or  x  A, Q(x) and  x  B, P(x). A is a proper subset of B if  x  A, Q(x) and  x  B,  P(x).

Disjoint Sets Sets A and B are disjoint if A  B = . A collection of sets A 1, A 2, …, A n are mutually disjoint, or pairwise disjoint, if A i  A j =  for all i and j, with i  j.

Examples The following sets are mutually disjoint. {0} {1, 2, 3, …} = N + {-1, -2, -3, …} = N - The following sets are mutually disjoint. {…, -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 {A 1, A 2, …, A n } is a partition of a set A if A 1, A 2, …, A n are mutually disjoint, and A 1  A 2  …  A n = A.

Examples {{0}, {1, 2, 3, …}, {-1, -2, -3, …}} is a partition of Z. {{…, -3, 0, 3, 6, …}, {…, -2, 1, 4, 7, …}, {…, -1, 2, 5, 11, …}} 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 A i = {n  N | f(n) = i}. Then A 0, A 1, A 2, … is a partition of N (except that it is infinite). Verify that A i  A j =  for all i, j, with i  j. A 0  A 1  A 2  … = 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 (  ))? If A contains n elements, how many elements are in P (A)? Prove it.

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 2 n subsets of U. Given an register of n bits, there are 2 n 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 = {a 0, a 1, a 2, …, a n – 1 }. Using a 32-bit integer to represent a set S, let bit i represent the element a i. If i = 0, then a i  S. If i = 1, then a i  S. For example, represents the set S = {a 0, a 2, a 3, a 4, a 7 }.

Example: SetDemo.cpp