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 true or false 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 ¬(P 1 V … V P n ) = (¬P 1 ) Λ … Λ (¬P n ) ¬ (P 1 Λ … Λ P n ) = (¬P 1 ) V … V (¬P n )

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? 1.DB models aspects of the real world 2.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 ( s), Gigabytes Internet ( s), 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

Example Customer table accountIdLast N.First N. 101BlockJane 102HamiltonCherry 103HarrisonKate 104BreauxCarroll Logical description (Schema) Customer (accountId, lastName, firstName) Table creation statement create table Customer (accountId integer, lastName char(20), firstName char(20))

Client/Server Interaction