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

Ac = {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 defineA = {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

Ac = {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), orx A, Q(x).

A equals B if x U, P(x) Q(x), orx 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 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}{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 {A1, A2, …, An} is a partition of a set A ifA1, A2, …, An are mutually disjoint, and

A1 A2 … An = 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 Ai = {n N | f(n) = i}.

Then A0, A1, A2, … is a partition of N (except that it is infinite).

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())? 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 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: SetDemo.cpp