basic definitions of set theory lecture 23 section 5.1 wed, mar 8, 2006
TRANSCRIPT
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