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Understanding Sets and Set Operations: Definitions, Membership, and Cartesian Products

This lecture focuses on the fundamentals of set theory, including definitions of sets, set membership, and various set operations. It discusses the characteristics of sets, such as unordered collections, the cardinality of sets, and the concept of power sets. The Cartesian product is explained with a focus on ordered pairs. Additionally, the lecture covers the significance of set notation, quantifiers, and Venn diagrams in illustrating the relationships between sets. By the end, learners will have a thorough understanding of basic set concepts necessary for further exploration in mathematics and its applications.

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Understanding Sets and Set Operations: Definitions, Membership, and Cartesian Products

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Presentation Transcript

  1. Lecture 6 2.1 Sets 2.2 Set Operations

  2. Definition of Set and Set Theory

  3. Describing Set Membership

  4. Set Builder Notation

  5. Sets and Set Operations Definition: A set is any collection of distinct things considered as a whole. A set is an unordered collection of objects. Discuss whether each of the following in a set: S = {1, 2, 3, 42} V = {x|x is a real number} T = {1, 1, 2, 3} W = {x|x is not in W} U = { } Z = {{1,2,3},{2,3,4},{3,4,5}} P = { { }, { { } }, { { { } } } } Q = {{1,2,3}, {2,3,4},{3,2,1}} Since the members of a set are in no particular order Q is not a set if its members are sets, but Q is a set if its members are 3-tuples, vectors or some other entity for which membership order is important. Since f = { } we can rewrite P = { f, {f}, {{f}} } so P is a set containing three elements, namely the empty set, a singleton containing the empty set and a singleton containing a singleton containing the empty set.

  6. If A is a set containing n elements then |A| = n, and is called the cardinality of A. Given a set S, the power set of S is the set of all subsets of the set S. The power set is S is denoted by P(S). The ordered n-tuple (a1, a2, . . . , an) is the ordered collection that has a1 as its first element, a2 as its second element . . . and an as its nth element. Let A and B be sets. The Cartesian product of A and B, denoted by AxB, is the set of all ordered pairs (a,b) where a A and b B. Hence, AxB = {(a,b)|a A b B}. Definitions: Set Properties

  7. B 2 4 5 1 1,2 1,4 1,5 3 3,2 3,4 3,5 5 5,2 5,4 5,5 8 8,2 8,4 8,5 A A = { s, pass, link, stock } B = { word, port, age, able} Cartesian Product A = { 1,3,5,8 } B = { 2,4,5 }

  8. Definition: Venn Diagrams

  9. Set Notation with Quantifiers For all x, elements of the Reals, x2 is greater than or equal to 0. There exists an x, element of the Integers, such that x2 equals 1. For every x, element of the Reals, there exists a y, element of the Reals, such that x times y = 1. (give an exception to show this statement is false) For every x, element of the Integers, there exists ay, element of the Integers, such that x plus y = 0.

  10. Truth Sets of Quantifiers

  11. Combining Sets

  12. Set Union

  13. Set Intersection

  14. A B U U U U U A B A B A B A B U U U U A B A B A B A B U U U U A B A B A B A B Venn Diagrams

  15. Set Identities (This is why we had a separate test on first-order logic.)

  16. 1 1 1 1 1 1 1 1 1 1 0 1 1 1 0 1 1 0 1 1 1 0 0 1 1 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 Membership Table Show that

  17. Computer Representation of Sets

  18. An Example Satisfiability Set Enumeration =

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