Relational Database Systems 1

Transcription

1 Relational Database Systems 1 Wolf-Tilo Balke Joachim Selke Institut für Informationssysteme Technische Universität Braunschweig

2 Exercise 4.1 The four major steps of conceptual schema integration 1. Pre-integration analysis Decide on the right integration strategy Result: Conflicting schemas and integration strategy 2. Comparison of schemas Identify conflicts in the schemas (naming/structure) Result: List of conflicts 3. Conformation of schemas Resolve the conflicts between the different schemas Result: Modified, non-conflicting schemas Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 2

3 Exercise 4.1 The four major steps of conceptual schema integration 4. Merging and restructuring Combine the modified schemas and do some optimization Result: Your new integrated global schema Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 3

4 Exercise 4.2 The three main goals of restructuring Completeness All different concepts of each individual schema are included in the global schema Minimality There is no redundant information in the global schema Ease of understanding The new model is easy to understand That is, for example: Entities are clearly distinguishable, the level of abstraction looks natural, Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 4

5 Exercise 4.3 Two models: A. The data of interest is about books. Books have titles. They are published by publishers with names and addresses. Books are adopted by universities having a name and belonging to a state. Books refer to certain topics. B. The data of interest includes publications of different types. Each publication has a title, a publisher, and a list of keywords. Each keyword consists of a name, a code, and a research area. Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 5

6 Exercise 4.3 Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 6

7 Exercise 4.3 Turn Keywords into Topics. Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 7

8 Exercise 4.3 Let Publisher become an entity. Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 8

9 Exercise 4.3 Put them together. Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 9

10 Exercise 4.3 Introduce a subset relationship. Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 10

11 Exercise 4.3 Drop duplicated properties from Book. Pictures taken from Bartini/Lenzerini/Navathe: A Comparative Analysis of Methodologies for Database Schema Integration (1987) Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 11

12 Exercise 4.4 Integrate that! EXAM DEPARTMENT Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 12

13 Exercise 4.4 Integrate that! STAFF PHONE STUDENT PHONE Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 13

14 Exercise 4.4 One solution of many: Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 14

15 Overview Basic set theory Relational data model Relational Algebra Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 15

16 5.1 Basic Set Theory Set theory is the foundation of mathematics You probably all know these things from your math course, but repeating never hurts The relational model is based on set theory; understanding the basic math will help a lot Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 16

17 5.1 Sets A set is a mathematical primitive, and thus has no formal definition A set is a collection of objects (called members or elements of the set) Objects (or entities) have to be understood in a very broad sense, can be anything from physical objects, people, abstract concepts, other sets, Objects belong (or do not belong) to a set (alternatively, are or are not in the set) A set consists of all its elements Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 17

18 5.1 Sets Sets can be specified extensionally List all its elements Example: A = {IfIS, 42, Balke, Hurz!} Sets can be specified intensionally Provide a criterion deciding whether an object belongs to the set or not (membership criterion) Examples: A = {x x > 4 and x Z} B = {x N x < 7} C = {all facts about databases you should know} Sets can be either finite or infinite Set of all super villains is finite Set of all numbers is infinite Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 18

19 5.1 Sets Sets are different, iff they have different members {a, b, c} = {b, c, a} Duplicates are not supported in standard set theory {a, a, b, c} = {a, b, c} Sets can be empty (written as {} or ) Notations for set membership: a {a, b, c} e {a, b, c} Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 19

20 5.1 Sets Defining a set by its intension: Intension must be well-defined and unambiguous There always is a clear membership criterion to determine whether an object belongs to the set (or not) Not a valid definition (Russell s paradox): In a small town, there is just one male barber. He shaves all and only those men in town who do not shave themselves. Does the barber shave himself? Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 20

21 5.1 Sets Still, the set s extension might be unknown (however, there is one) Example: All students in this room who are older than 22. Well-defined, but not known to us But (at least in principle) we can find out! As we will see later: Intension Database query Extension Result of a query Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 21

22 5.1 Sets For every set, there is an accompanying definition of equality (or equivalence) Is x = y? If they are equal, they are actually just one element However, you could have two different descriptions of the same element Example: The set of all 26 standard letters ö is not contained in this set m = M and is also element of the set ( m and M are different descriptions of the same object) Example: The set of all 59 letters and umlauts in German ö is element of the set m M and both are element of the set ( m and M two different objects) Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 22

23 5.1 Sets Sets have a cardinality (i.e., number of elements) Denoted by A {a, b, c} = 3 A B B A Set A is a subset of set B, denoted by A B, iff every member of A is also a member of B B is a superset of A, denoted by B A, iffa B Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 23

24 5.1 Tuples A tuple (or vector) is a sequence of objects Length 1: Singleton Length 2: Pair Length 3: Triple Length n: n-tuple In contrast to sets Tuples can contain an object more than once The objects appear in a certain order The length of the tuple is finite Written as a, b, c or (a, b, c) Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 24

25 5.1 Tuples Hence: <a, b, c> <c, b, a>, whereas {a, b, c} = {c, b, a} <a 1, a 2 > = <b 1, b 2 > iff a 1 =b 1 anda 2 =b 2 n-tuples (n > 2) can also be defined as a cascade of ordered pairs: <a, b, c, d> = <a, <b, <c, d>>> Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 25

26 5.1 Set Operations Four binary set operations Union, intersection, difference and Cartesian product Union: Creates a new set containing all elements that are contained in (at least) one of two sets {a, b} {b, c} = {a, b, c} Intersection: Creates a new set containing all elements that are contained in both sets {a, b} {b, c} = {b} A B A B A A B B Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 26

27 5.1 Set Operations Difference: Creates a set containing all elements of the first set without those also being in the second set {a, b} {b, c} = {a} A A B B Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 27

28 5.1 Set Operations Cartesian product: The Cartesian product is an operation between two sets, creating a new set of pairs such that: A B = {<a, b> a A andb B} Named after René Descartes Example: {a, b} {b, c} = {<a, b>, <a, c>, <b, b>, <b, c>} Cleverness = { genius, dumb } Character = { hero, villain } Cleverness Character = { <genius, hero>, <dumb, hero>, <genius, villain>, < dumb, villain> } The Cartesian product can easily be extended to higher dimensionalities. Example: A B C is a set of triples. Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 28

29 5.1 Relations A relationrover some sets D 1,, D n is a subset of their Cartesian product R D 1 D n The elements of a relation are tuples The D i are called domains Each D i corresponds to an attribute of a tuple n=1: Unary relation or property n=2: Binary relation n=3: Ternary relation Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 29

30 5.1 Relations Some important properties: Relations are sets in the mathematical sense, thus no duplicate tuples are allowed The list of tuples is unordered The list of domains is ordered Relations can be modified by inserting new tuples, deleting existing tuples, and updating (that is, modifying) existing tuples. Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 30

31 5.1 Relations A special case: Binary relations R D 1 D 2 D 1 is called domain,d 2 is called codomain (range, target) Relates objects of two different sets to each other Ris just a set of ordered pairs R = {<a,1>, <c,1>, <d,4>, <e,5>, <e,6>} Can also be written as ar1, cr1, dr4, Imagine Likes Person Beverage Joachim Likes Coffee, Tilo Likes Tea, D b e c a R 1 2 d V Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 31

32 5.1 Relations Example: accessory = {spikes, butterfly helmet} material = {silk, armor plates} color = {pink, black} color material accessory = {<pink, silk, butterfly helmet>, <pink, silk, spikes>, <pink, armor plates, butterfly helmet>, <pink, armor plates, spikes>, <black, silk, butterfly helmet>, <black, silk, spikes>, <black, armor plates, butterfly helmet>, <black, armor plates, spikes>} Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 32

33 5.1 Relations Relation FamousHeroCostumes color material accessory FamousHeroCostumes = {<pink, silk, butterfly helmet>, <black, armor plates, spikes>} Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 33

34 5.1 Functions Functions are special case of binary relations Partial function: Each element of the domain is related to at most one element in the codomain Total function: Each element in the domain is related to exactly one element in the codomain Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 34

35 5.1 Functions Functions can be used to abstract from the exact order of domains in a relation Alternative definition of relations: A relation is a set of functions. Every tuple in the relation is considered as a function of the type {A 1,,A n } D 1 D n That is, every tuple maps each attribute to some value Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 35

36 5.1 Functions Example: color = {pink, black} material = {silk, armor plates} accessory = {spikes, butterfly helmet} The tuple <pink, silk, butterfly helmet> can also be represented as the following function t: t(color) = pink t(material) = silk t(accessory) = butterfly helmet Usually, one writes t[color] instead of t(color) Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 36

37 5.2 Relational Model Well, that s all nice to know But: We are here to learn about databases! Where is the connection? Here it is A data model is a description of concepts in terms of attributes and domains A database instance is a set of objects having certain attribute values Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 37

38 5.2 Relational Model OK, then Giving a data model (e.g., by ER modeling) determines entities and relationships, as well their corresponding sets of attributes and associated domains The Cartesian product of the respective domains is the set of all possible instances (of each entity type or relationship type) A relation formalizes the actually existing subset of all possible instances Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 38

39 5.2 Relational Model Data models are described by relation schemas, denoted by R(A 1 :D 1,, A n :D n ) The actual database instance is given by a set of matching relations Example Relation schema: CATS(name : varchar(10), age : integer) A matching relation: { (Blackie, 10), (Pussy, 5), (Fluffy, 12) } Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 39

40 5.2 Relational Model Relations can be written as tables: relation name attributes tuples PERSON firstname lastname sex Clark Joseph Kent m Louise Lane f Lex Luthor m Charles Xavier m Erik Magnus m Jeanne Gray f Ororo Munroe f Tony Edward Stark m Matt Murdock m Raven Wagner f Robert Bruce Banner m domain values Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 40

41 5.2 Relational Model A relational database schema consists of a set of relation schemas a set of integrity constraints A relational database instance (or state) is A set of relations adhering to the respective schemas and respecting all integrity constraints Relation Schemas R a b c x 67 zv Integrity Constraints y 56 qa Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 41

42 5.2 Relational Model Every relational DBMS needs a language to define its relation schemas (and integrity constraints) Data definition language (DDL) Typically, it is difficult to formalize all possible integrity constraints, since they tend to be complex and vague A relational DBMS also needs a language to handle tuples Data manipulation language (DML) Today s RDBMS use SQL as DDL and DML Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 42

43 5.2 Conversion from ER After modeling a conceptual schema (e.g., using an ER diagram), the schema can be automatically transformed into a relational schema Remember: conceptual design ER diagram UML, logical design tables, columns, physical design tablespaces, Indexes, Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 43

44 5.2 Conversion from ER Each entity typeewith attributes A 1,, A n from domainsd 1,, D n is converted into an n-ary relation schemae(a 1 :D 1,, A n :D n ) If there is a relationship type E is_af involved (specialization), the inheritance relationship can be expressed by copying all key attributes from F Person telephoneno weakness alias Hero firstname lastname Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 44

45 5.2 Conversion from ER A relationship type R between entity types E 1,, E n is converted to a relation schema whose attributes are all the key attributes of E i If keys share the same name, they have to be renamed If the relationship type has own attributes they are also copied to the relation schema Hero uses Power situation Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 45

46 5.2 Conversion from ER Example: Person telephoneno firstname lastname Hero uses Power reach weakness alias situation type name Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 46

47 5.2 Conversion from ER Entity types: Person telephoneno firstname lastname Hero Person(firstname:Varchar(20), lastname: Varchar(30), telephoneno: Integer) alias weakness Hero(firstname: Varchar(20), lastname: Varchar(30), alias : Varchar(20), weakness : Varchar(10)) Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 47

48 5.2 Conversion from ER Relationship types: Hero uses Power reach weakness alias situation type name Hero(firstname: Varchar(20), lastname: Varchar(30), alias : Varchar(20), weakness : Varchar(10)) Power(name :Varchar(20), type : Varchar(20), reach : Integer) Uses(firstname:Varchar(20), lastname: Varchar(30), name : Varchar(20), situation : Varchar(30)) Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 48

49 5.2 Conversion from ER Note: The ER diagram is semantically richer than the relational model Examples: No key constraints and functionalities yet Integrity constraints like disjoint/overlapping generalization cannot be expressed Therefore, it usually is a really good idea to create an ER diagram before coding a logical schema Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 49

50 5.2 Integrity Constraints Integrity constraints are difficult to model in ER Basically annotations to the diagram, especially for behavioral constraints Example: The popularity rating of any assistant should always be less than the respective professor s But some structural constraints can directly be expressed Key constraints Functionalities Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 50

51 5.2 Inherent Constraints A relation is defined as a set of tuples All tuples have to be distinct, i.e., no two tuples can have the same combinations of values for all attributes So-called uniqueness (unique key) constraint Any subset of a relation type s attributes is called a superkey if two tuples never can share the same values with respect to this subset The set of all attributes is always a superkey, but there may be smaller sets Superkeys may have redundant attributes Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 51

52 5.2 Inherent Constraints A minimal superkey is called a key Minimal means that no attribute can be removed without losing the superkey property Of course, a relation can have several keys The key property is determined from the semantics of the attributes, not from the current data instances Example: Relation address(street, number, zip code, city) Keys: {street, number, zip code} and {street, number, city} Assumption: No city contains streets that share the same name but differ only in their zip code Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 52

53 5.2 Inherent Constraints The set of keys of some relation R is called candidate key set or cand(r) For each relation, a single candidate key has to be chosen to identify tuples in the relation, the so-called primary key Analogously to ER diagrams, the chosen primary key is often underlined in the relation schema Example: address(street, number, zip-code, city) Though any candidate key can be chosen, it is usually better to choose a primary key with a small number of attributes Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 53

54 5.2 Inherent Constraints Assume that an ER diagram is converted into relation schemas, what are the candidate keys of the relation schemas? If the relation schema has been derived from some entity type E with key attributes K E := {A 1,, A n }, then K E cand(r) has been derived from an N:M relationship type betweeneandf, thenk E K F cand(r) Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 54

55 5.2 Inherent Constraints If the relation schema has been derived from an 1:1 relationship type between E and F, then cand(e) cand(f) = cand(r) has been derived from an N:1 relationship type betweene 1,,E n andf, thenk E 1 K E n cand(r) In this case, it might also be a good to add F s key attributes Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 55

56 5.2 Inherent Constraints Another constraint from ER diagrams is whether a value has to be provided for some attribute NULL values, allowed by default Again, this is a semantic property A second inherent constraint for each relation is that primary keys must never be NULL So-called entity integrity constraint Example: address( street NOT NULL, number NOT NULL, zip-code, city NOT NULL ) Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 56

57 5.2 Inherent Constraints A third integrity constraint applies to all relation schemas that have been derived from relationship types Relationship types borrow their (primary) keys from the entities involved, so-called foreign keys Relationships only exists, if the respective entities exist So-called referential integrity constraint Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 57

58 5.2 Inherent Constraints Example: Hero uses Power situation Uses(firstname:Varchar(20), lastname: Varchar(30), name: Varchar(20), situation : Varchar(30)) Foreign key from Hero Foreign key from Power If a tuple ( Clark, Kent, X-Ray vision, Bomb threat ) exists in Uses, there has to be a tuple ( Clark, Kent, ) in Hero and a tuple ( X-Ray vision, ) in Power Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 58

59 5.2 Inherent Constraints All these three structural constraints have to be checked by the database Unique key constraint Entity integrity constraint Referential integrity constraint This is especially necessary when inserting, deleting, or updating tuples in relations Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 59

60 5.2 First Normal Form There is a another major constraint on the attributes data types in the relational model The value of any attribute must be atomic, that is, it cannot be composed of several other attributes If this property is met, the relation is often referred to as a being in first normal form (1NF or minimal form) In particular, set-valued and relation-valued attributes (tables within tables) are prohibited Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 60

61 5.2 First Normal Form Example of a set-valued column: A person may own several telephones (home, office, cell, ) Person firstname lastname telephoneno Clark Joseph Kent Louise Lane { , , } Lex Luthor Charles Xavier Erik Magnus { , } Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 61

62 5.2 First Normal Form Note: It is possible to model composed attributes in ER models To transform such a model into the relational model, a normalization step is needed This is not always trivial, e.g., what happens to keys? Person firstname lastname telephoneno Clark Joseph Kent Louise Lane Louise Lane Louise Lane Lex Luthor Charles Xavier Erik Magnus Erik Magnus Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 62

63 5.2 First Normal Form In a purely relational database, all relations are in first normal form Object-oriented databases feature multi-valued attributes, thus closing the modeling gap Object-relational extensions integrate user-defined types (UDTs) into relational databases Oracle from version 9i, IBM DB2 from version 8.1, Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 63

64 5.3 Relational Algebra How do you work with relations? Relational algebra! Proposed by Edgar F. Codd: A Relational Model for Large Shared Data Banks, Communications of the ACM, 1970 The theoretical foundation of all relational databases Describes how to manipulate relations and retrieve interesting parts of available relations Relational algebra is mandatory for advanced tasks like query optimization Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 64

65 5.3 Relational Algebra Elementary operations: Set algebra operations Set Union Set Intersection Set Difference Cartesian Product New relational algebra operations Selection σ Projection π Renaming ρ Additional derived operations (for convenience) All sorts of joins,,, Division EN 6 Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 65

66 5.3 Example Relations students matnr firstname lastname sex 1005 Clark Kent m 2832 Louise Lane f 4512 Lex Luther m 5119 Charles Xavier m 6676 Erik Magnus m 8024 Jeanne Gray f 9876 Logan m courses crsnr title 100 Intro. to being a Superhero 101 Secret Identities How to take over the world exams student course result EN 6 Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 66

67 5.3 Relational Algebra Selection σ Selects all tuples (rows) from a relation that satisfy some given Boolean predicate (condition) Selection = Create a new relation that contains exactly the satisfying tuples σ <condition condition>(r) Condition clauses: <attribute> θ<value> <attribute> θ<attribute> θ {=, <,,, >, } Clauses may be connected by, and (logical AND, OR, and NOT) EN 6 Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 67

68 5.3 Relational Algebra Example: students matnr firstname lastname sex 1005 Clark Kent m 2832 Louise Lane f 4512 Lex Luther m 5119 Charles Xavier m 6676 Erik Magnus m 8024 Jeanne Gray f 9876 Logan m Select all female students σ sex= f (students) EN 6 σ sex= f (students) matnr firstname lastname sex 2832 Louise Lane f 8024 Jeanne Gray f Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 68

69 5.3 Relational Algebra Example: exams student course result Select exams within course 100 with a result of worse than 3.0 σ course=100 result>3.0 (exams) σ course=100 result>3.0 (exams) student course result EN 6 Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 69

70 5.3 Relational Algebra Projection π Retains only attributes (columns) with given names Again: Creates a new relation with only those attribute sets which are specified within some attribute list If tuples are identical after projection, duplicate tuples are removed π <attributelist> (R) courses crsnr title 100 Intro. to being a Superhero 101 Secret Identities How to take over the world All courses titles : π title (courses) π title (courses) title title Intro. to being a Superhero Secret Identities 2 How to take over the world EN 6 Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 70

71 5.3 Relational Algebra Of course, these operations can be combined Retrieve first name and last name of all female students π firstname, lastname sex=ʼfʼ students firstname lastnameσ sex ʼfʼ students matnr firstname lastname sex 1005 Clark Kent m 2832 Louise Lane f 4512 Lex Luther m 5119 Charles Xavier m 6676 Erik Magnus m 8024 Jeanne Gray f 9876 Logan m σ sex= f students matnr firstname lastname sex 2832 Louise Lane f 8024 Jeanne Gray f π firstname, lastname σ sex=ʼfʼstudents firstname lastname Loise Lane Jeanne Gray EN 6 Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 71

72 5.3 Relational Algebra Renaming operator ρ Renames a relation and/or its attributes Also denoted by ρ S(B1, B2,, Bn) (R) or ρ S (R) or ρ (B1, B2,, Bn) (R) courses Retrieve all course numbers contained in courses and name the resulting relation lectures lectures π crsnr courses crsnr crsnr title 100 Intro. to being a Superhero 101 Secret Identities How to take over the world lectures crsnr EN 6 Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 72

73 5.3 Relational Algebra Renaming operator ρ Select all result of course 100 from exams; the resulting relation should be called result and contain the attributes matno, crsno, and grade ρ results(matno matno, crsno, grade)σ course=100 exams results matno crsno, grade) course =100 σ course=100 exams student course result results matno crsno grade EN 6 Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 73

74 5.3 Relational Algebra Union, intersection, and set difference Operators work as in set theory Operands have to be union-compatible (i.e., they must consist of the same attributes) Written as R S, R S, and R S, respectively σ course=100 exams course title σ course=100 exams σ course=102 exams 100 Intro. to being a Superhero course title σ course=102 exams course title 100 Intro. to being a Superhero 102 How to take over the world 102 How to take over the world EN 6 Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 74

75 5.3 Relational Algebra Cartesian product Written as R S Also called cross product Creates a new relation by combining each tuple of the first relation with every tuple of the second relation Attribute set = all attributes of R plus all attributes of S The resulting relation contains exactly R S tuples EN 6 Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 75

76 5.3 Relational Algebra goodgrades females student course result matno lastname Lane Gray cross student course result matno lastname Lane Lane Lane Gray Gray Gray females π matno goodgrades σ result 3.0 exams result 3.0 matno, lastnameσ sex=female femalestudents cross goodgrades females EN 6 Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 76

77 5.3 Relational Algebra π lastname, title, result σ matno=student course=crsno crsno crsno females goodgrades courses lastname course result Lane How to take over the world? 2.0 This type of statement is very important! Cartesian product, followed by selections/projections Selections/projections involve different source relations This kind of query is called a join EN 6 Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 77

78 5.3 Relational Algebra Theta join (θ) (θ is a Boolean condition) Sometimes also called inner join or just join Creates a new relation by combining related tuples from different source relations Written as R σ(condition condition) S or Theta joins are very similar to selections π lastname, title, result result females σ(matno matno=student π lastname, title, result σ matno=student course=crsno crsno lastname course result Lane How to take over the world? 2.0 σ (condition condition)(r S) student)goodgrades σ(course course=crsno crsno) courses crsno(females goodgrades courses) EN 6 Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 78

79 5.3 Relational Algebra Equi-join (condition condition) Joins two relations only using equality conditions R (condition condition) S condition may only contain equality statements between attributes (A 1 =A 2 ) A special case of the theta join π lastname, title, result result females matno student goodgrades course matno=student π lastname, title, result σ matno=student course=crsno crsno lastname course result Lane How to take over the world? 2.0 course=crsno crsno crsno courses crsno females goodgrades courses EN 6 Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 79

80 5.3 Relational Algebra Natural join A special case of the equi-join R (attributelist attributelist) S Implicit join condition Each attribute in attributelist must be contained in both source relations For each of these attributes, an equality condition is created All these conditions are connected by logical AND If attributelist is empty: Join attributes = All attributes that are shared between the two relations (that is, that have the same name) EN 6 Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 80

81 5.3 Relational Algebra goodgrades matno crsnr result females matno lastname 2832 Lane 8024 Gray courses crsnr title 100 Intro. to being a Superhero 101 Secret Identities How to take over the world π lastname, title, result females goodgrades courses lastname course result Lane How to take over the world? 2.0 EN 6 Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 81

82 5.3 Relational Algebra Left outer join and right outer join Theta-joins and its specializations join exactly those tuples that satisfy the join condition Tuples without a matching join partner are eliminated and will not appear in the result All information about non-matching tuples gets lost Outer joins allow to keep all tuples of a relation Padding with NULL values if there is no join partner Left outer join : Keeps all tuples of the left relation Right outer join : Keeps all tuples of the right relation Full outer join : Keeps all tuples of both relations EN 6 Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 82

83 5.3 Relational Algebra Example: List students and their exam results π lastname, crsnr, result result students exams students exams matnr firstname lastname sex matnr crsnr result 1005 Clark Kent m Louise Lane f Lex Luther m Charles Xavier m π lastname, crsnr, result students exams lastname crsnr result Kent Kent Lane Lex Luther and Charles Xavier are lost because they didn t take any exams! Also, information on student 9876 disappears Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 83

84 5.3 Relational Algebra Left outer join: List students and their exam results π lastname, crsnr, result students matnr firstname lastname sex 1005 Clark Kent m 2832 Louise Lane f 4512 Lex Luther m 5119 Charles Xavier m π lastname, crsnr, result students exams lastname crsnr result Kent Kent Lane Luther NULL NULL Xavier NULL NULL result students exams exams matnr crsnr result Now, you could easily count all non-null results to get student statistics without looking into the students table. Kent: 2; Lane: 1; Luther: 0; Xavier: 0 Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 84

85 5.3 Relational Algebra Right outer join: π lastname, crsnr, result students result students exams exams matnr firstname lastname sex matnr crsnr result 1005 Clark Kent m Louise Lane f Lex Luther m Charles Xavier m π lastname, crsnr, result students exams lastname crsnr result Kent Kent Lane NULL All exam result with student names if known. Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 85

86 5.3 Relational Algebra Full outer join: π lastname, crsnr, result students matnr firstname lastname sex 1005 Clark Kent m 2832 Louise Lane f 4512 Lex Luther m 5119 Charles Xavier m lastname crsnr result Kent Kent Lane Luther NULL NULL NULL Xavier NULL NULL result students exams exams matnr crsnr result π lastname, crsnr, result students exams Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 86

87 5.3 Relational Algebra Left semi-join and right semi-join Combination of a theta-join and projection R (condition S π (list of all attributes in R) (R) (condition R (condition S π (list of all attributes in S) (R) (condition condition)s condition S condition)s condition S (list of all attributes in S) condition)s condition)s condition Creates a copy of R (or S) while removing all those tuples that do not have a join partner A filtering operation Works like a natural join if condition is empty Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 87

88 5.3 Relational Algebra Left semi-join: students exams students matnr firstname lastname sex 1005 Clark Kent m 2832 Louise Lane f 4512 Lex Luther m 5119 Charles Xavier m exams matnr crsnr result students exams matnr firstname lastname sex 1005 Clark Kent m 2832 Louise Lane f All hard-trying students (those who did exams). Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 88

89 5.3 Relational Algebra Division Written R S S contains only attributes that are also present in R Division restricts R in terms of attributes and tuples Result s attributes = those in R but not in S Result s tuples = those in R such that any tuple in S is a join partner Useful for queries like Find the sailors who have reserved all boats. More formal: Let A R ={attributes of R}; A S ={attributes of S}; A S A R A = A R A S R S π A (R) π A ((π A (R) S) R R) Why the name division? Because (R S) S) S S = R. But: (R S) S S = R is not true in general Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 89

90 5.3 Relational Algebra Division: SC (= students and courses) matnr name crsnr 1000 Clark Kent C Clark Kent C Louise Lane C Lex Luther C Lex Luther C Lex Luther C Charles Xavier C Charles Xavier C100 Result contains all those students who took the same courses as Clark Kent (and possibly some more). CCK (= courses of Clark Kent) CCK π crsnr σ name= Clark Kent SC crsnr C100 C102 SC CCK matnr name 1000 Clark Kent 1002 Lex Luther EN 6 Relational Database Systems 1 Wolf-Tilo Balke Institut für Informationssysteme TU Braunschweig 90