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Basic Definitions of Set Theory

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 =  .

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Basic Definitions of Set Theory

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  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} • {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}

  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. • {{…, -3, 0, 3, 6, …}, {…, -2, 1, 4, 7, …}, {…, -1, 2, 5, 11, …}} 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. 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, 10011101 represents the set S = {a0, a2, a3, a4, a7}.

  12. Example: Sets.cpp

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