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Sets: Definitions, Operations, and Partitioning

Learn about the basics of sets, including their definitions, operations, and partitioning. Understand universal and empty sets, set negation, and set diagrams. Explore partitions and cross partitions, and the importance of context within a universe of discourse.

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Sets: Definitions, Operations, and Partitioning

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  1. Chapter 4 Sets

  2. Sets • A set is a well-defined collection of objects. • There are two ways to define a set: (1) by listing all the members of the set, and (2) by giving a rule for determining whether objects do or do not belong to the set. Call (1) a “list” definition and (2) a “rule” definition. • For example, suppose we study the relation between voting behavior and political preference. Political preference can be defined as being a registered Republican or Democrat. The rule is usually sufficient. Such a rule might be: A Republican is any person who is registered with the Republican Party.

  3. Subsets • A subset of a set is a set that results from selecting sets from an original set. The set B is a subject of a set A whenever all the elements of B are elements of A. We write • Examples are numerous: religious preferences into Protestants, Catholic, Jew; intelligence into high and low.

  4. Operations • There are two basic set operations: intersection and union. For intersection the logical operator is “and.” For union, the proper logical operators is “or.” • The symbol for intersection is (read “intersection” or “cap”) • The union of two sets is written

  5. The Universal and Empty Sets; Set Negation • The universal set, labeled U, is the set of all elements under discussion. • The empty set is the set with no members in it. We label it E • The negation or complement of the set A is written ~A.

  6. Set Diagrams • Figure 4.2~4.6

  7. Partitions and Cross Partitions • U can be broken down (partitioned) into subsets that do not intersect and that exhaust all of U. When this is done the process is called partitioning. • Formally stated, partitioning breaks down a universal set into subsets that are disjoint and exhaustive of the universal set. • [A1 A2] and [B1 B2] are partitions if: • and • and

  8. Partitions and Cross Partitions • It is possible to put the two partitions together into a cross partition. A cross partition is a new partitioning that arises from successively partitioning the same set U by forming all subsets of the form • Figure 4.8

  9. Levels of Discourse • When we talk about anything we talk about it in a context or frame of reference. The expression, context and frame of reference are closely related to U, the universe of discourse. The universe of discourse must be able to include any objects we talk about. • Research requires precise definitions of universal sets. Precise means to give a clear rule that tells you when an object is or is not a member of U. • Figure 4.9-4.10

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