1 / 23

Basic Definitions of Set Theory

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 .

brooke
Download Presentation

Basic Definitions of Set Theory

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

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

  2. 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.

  3. 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}.

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

  5. Set Differences • Define the differenceAminusB 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).

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

  7. Subsets • A is a subset of B, written A B, if x  A, x  B. • AequalsB, 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.

  8. 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).

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

  10. 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)}.

  11. Complements and Differences • The complement of A is Ac = {x U | P(x)}. • The differenceAminusB 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)}.

  12. 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).

  13. 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.

  14. 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}

  15. 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.

  16. 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.

  17. 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.

  18. 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.

  19. 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.

  20. 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}?

  21. 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.

  22. 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}.

  23. Example: SetDemo.cpp

More Related