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## Sets and Set Operations

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**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**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**Sets and Set Operations**• Set operations: Union A B Intersection A B Difference A B Complement (Universe of discourse)A • Venn diagram**Useful relations**• | A B | = |A| + |B| - |A B| • DeMorgan’s law co(A B) = co(A) co(B) • DeMorgan’s law can be generalized**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**Propositions**• DeMorgan’s laws ¬(P1V …V Pn ) = (¬P1) Λ … Λ(¬Pn) ¬ (P1Λ … ΛPn ) = (¬P1)V … V (¬Pn)**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**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**Uses of Databases**• Traditional (airline reservation, school, hospital, …) • Online Shopping • Search Engines • Genetic Databases • Geographical Information Systems**Sizes of Database**• Personal (1 User), Megabytes • Workgroup (<25 Users), Megabytes • Organizational (100-1000s), Gigabytes • Internet (100-1000s), Terabytes**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)**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