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Sets and Set Operations

Sets and Set Operations

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Sets and Set Operations

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  1. Sets and Set Operations • A set is an unordered collection of objects called elements Explicit definition Implicit definition • Two sets A and B are equal if every element in A is also in B and every element in B is also in A (A = B) • A is contained in B if every element in A is also an element of B (A  B) Example: Φ A for every set A

  2. Sets and Set Operations • The power set of A, Ρ(A) is the set of all subsets of A Example: Ρ(Φ) = {Φ} • The cross product of two sets A x B = { (a, b) | a  A and b  B} order is important: A x B  B x A in general (x is not commutative) • A relation is a subset of A x B • cross product can be generalized

  3. Sets and Set Operations • Set operations: Union A  B Intersection A  B Difference A  B Complement (Universe of discourse)A • Venn diagram

  4. Useful relations • | A  B | = |A| + |B| - |A  B| • DeMorgan’s law co(A  B) = co(A)  co(B) • DeMorgan’s law can be generalized

  5. Propositions • A proposition P is a statement that is either trueorfalse Example P : It is raining today in Chicago • Compound propositions and their truth values P or Q (P V Q), P and Q (P Λ Q), not P (¬P , or P), P  Q, P Q

  6. Propositions • DeMorgan’s laws ¬(P1V …V Pn ) = (¬P1) Λ … Λ(¬Pn) ¬ (P1Λ … ΛPn ) = (¬P1)V … V (¬Pn)

  7. Databases • A database is a system that allows storage and manipulation of information • A database management system (DBMS) is a combination of hardware and software that allows organization and manipulation of the information

  8. Databases • Why are databases important? databases vs. Archiving, large size of data, quick access and manipulation • How and why are databases used? • DB models aspects of the real world • Collection of data • logically coherent • Meaningful • 3. Designed for specific purpose

  9. Uses of Databases • Traditional (airline reservation, school, hospital, …) • Online Shopping • Search Engines • Genetic Databases • Geographical Information Systems

  10. Sizes of Database • Personal (1 User), Megabytes • Workgroup (<25 Users), Megabytes • Organizational (100-1000s), Gigabytes • Internet (100-1000s), Terabytes

  11. How do database represent Info. • The physical database: • a collection of files containing the data content • The schema: • a specification of the physical database’s information content and logical structure • The database engine: • software that lets people access and modify the database contents • The data definition and manipulation languages: • SQL (Structured Query Language)

  12. Relational database manag. Systems (RDBMS) • RDBMS • Tables of data • Schema • Name of table • Names and types of attributes • Contents • Row is a fact • Attribute value is a characteristic

  13. Example

  14. Client/Server Interaction