1. Chapter 6 Relations

2. Topics in this Chapter Tuples Relation Types Relation Values Relation Variables SQL Facilities

3. Tuples A tuple is a set of ordered triples each containing an attribute name, attribute type, and a value Part# Part# P1 attribute name attribute type attribute value attribute triple

4. Tuples The count of ordered triples is the degree or arity of the tuple An attribute has a name and a type The set of attributes is the heading of the tuple Part# Part# Name Name Color Color etc. P1 Nut Red etc. attribute attribute value tuple heading

5. Tuples and their Properties Every tuple contains one value of the appropriate type for each attribute Attributes are unordered Every subset of a tuple is a tuple Tuples are of n-ary degree (based on attribute count) The empty tuple is nullary

6. Tuple Type Generation – General Form and Example General form TUPLE { }; Example VAR ADDR TUPLE { STREET CHAR, CITY CHAR, STATE CHAR, ZIP CHAR };

7. Tuple Type Generation – Selector Example Every tuple type has a selector operator Example TUPLE { STREET ‘808 Commonwealth Ave.’, CITY ‘Boston’, STATE ‘MA’, ZIP ‘02115’ };

8. Tuple Type Generation – Explanation of Selector Example The preceding tuple can be assigned to the tuple variable ADDR It can be tested for equality with any other tuple of the same type To be of the same type it is necessary and sufficient that they have the same attributes Tuples can have attributes of any type whatsoever, including relation types and tuple types

9. Tuple Operators - Equality The following well-known topics in relational database theory depend directly on this: Relational algebra Candidate keys Foreign keys Functional, and other, dependency

10. Tuple Operators – Equality – A Formal Definition Two tuples are equal if and only if they have the same attributes and for each attribute in one tuple, its value matches the value in the corresponding attribute in the other Two tuples are duplicates if and only if they are equal Two equal tuples are in fact the same tuple All nullary tuples are equal, so there is but one nullary tuple, forever and ever

11. Tuple Operators – Projection ADDR ( CITY, ZIP ) when operating against the previous tuple results in ADDR ( CITY ‘Boston’, ZIP ‘02115’) Tuple type inference implies that in any operation, such as the above projection, the types of the attributes adhere to the result TUPLE ( CITY CHAR, ZIP CHAR )

12. Tuple Operators – WRAP and UNWRAP – Two example types TUPLE { NAME NAME, ADDR TUPLE { STREET CHAR, CITY CHAR, STATE CHAR, ZIP CHAR } } Tuple type 1 (TT1) compare to TT2: TUPLE { NAME NAME, STREET CHAR, CITY CHAR, STATE CHAR, ZIP CHAR }

13. Tuple Operators – WRAP and UNWRAP – Discussion of the two example types Tuple type 1 (TT1) includes an attribute that is itself a tuple type It’s degree is two (TT2 is 5-ary) Let NADDR1 and NADDR2 be tuple variables of type TT1 and TT2, respectively

14. Tuple Operators – WRAP and UNWRAP – Operators at Work NADDR2 WRAP { STREET, CITY, STATE, ZIP } AS ADDR The result of the wrap is a tuple of TT1, so this assignment is valid: NADDR1 := NADDR2 WRAP { STREET, CITY, STATE, ZIP } AS ADDR; Analogously, this assignment is valid: NADDR2 := NADDR1 UNWRAP ADDR;

15. Relations – A Formal Definition A relation is a relation value, strictly speaking Every relation consists of a head and a body The heading of a relation is a tuple heading, i.e., the complete set of attributes, where every attribute has a name and a type The body of a relation is the set of tuples all having that heading The count of tuples is the cardinality of the relation

16. Relations – A Formal Definition, continued A relation does not contain tuples A relation contains a head and a body The body of the relation contains tuples The count of attributes is the degree (just as was the case with tuples) Every subset of a head is a head Every subset of a body is a body The empty head, or body, is a valid subset

17. Relations – A Relation Type Generator A type generator can be invoked in the definition of a relvar Takes the same form as the tuple type generator VAR PART_STRUCTURE RELATION {MAJOR_P# P#, MINOR_P#, P#, PQTY QTY} Comes with a selector operator

18. Relation Values – a/k/a Relations Every tuple contains exactly one value for each attribute There is no ordering of the attributes Likewise with the tuples There are no duplicate tuples Since every attribute of every tuple has a value, a relation is in first normal form, by definition

19. Relations vs. Tables A relation cannot have a duplicate tuple by definition; tables, if ill-managed, can Rows and columns are ordered, where tuples and attributes are not A table must be created with at least one attribute; a relation needs none Tables may contain nulls, a relation cannot Tables are “flat” or two-dimensional; relations are of n-dimensions

20. Relations vs. Tables A table may be said to represent a relation, if the table agrees to some stipulations: Each column will have an underlying type, and all atomic values will be of that type Row and column orderings will be irrelevant Duplicate rows are forbidden! “Then and only then may the table sip from the purifying stream of relational mathematics”

21. Relations-Valued Attributes Contrary to earlier reports, the death of repeating groups was greatly exaggerated It is perfectly relational to use relations as attributes, since a relation can be defined as a type Mr. Date humbly prostrates himself before the community, begs forgiveness, and promises it won’t happen again

22. Relations without Attributes – DUM and DEE The empty set is defined as the set with no members; it is a valid set A relation may have no attributes A relation with no attributes may have at most one tuple, the 0-tuple A relation with no attributes may, in the alternative, have no tuple at all TABLE_DEE is the one with the tuple; TABLE_DUM has none (Also known as “DEE” and “DUM”)

23. DUM n DEE They play a role in relational algebra akin to zero in arithmetic They are valid relations that do not map well to tables They are fundamentally important because DEE can mean “true” and DUM can mean “false”

24. Operators on Relations Selector, assignment, equality comparison Equals and not equals, equally Subset of, and superset of Proper subset, and proper superset, of Greater than and less than do not operate on relations

25. Operators on Relations – Continued IS_EMPTY Tuple is included in relation Extract tuple from a 1-relation Restrict, project, join For presentation purposes only, ORDER BY

26. Relation Variables – a/k/a Relvars Syntax for defining a base relvar: VAR BASE [ ]; where relation type takes the form RELATION { } which invokes the relation type generator

27. Relation Variables – an example – the Supplier VAR S BASE RELATION {S# S#, SNAME NAME, STATUS INTEGER CITY CHAR } PRIMARY KEY { S# };

28. Relation Variables – an example – the Part VAR S BASE RELATION {P# P#, PNAME NAME, COLOR WEIGHT CITY CHAR } PRIMARY KEY { P# };

29. Relation Variables – an example – the Supplier Part VAR SP BASE RELATION {S# S#, P# P#, QTY QTY,} PRIMARY KEY { S#, P# } FOREIGN KEY { S# } REFERENCES S FOREIGN KEY { P# } REFERENCES P;

30. Relation Variables – The S, P, SP example When a base relvar is defined, it is given an an initial value that is the empty relation of its relation type Relvars, like relations, define a predicate For a relvar it is the predicate that is common to all its possible values e.g. The supplier with supplier number S# is named SNAME, has a status STATUS, and is located in city CITY

31. Updating Relvars Update uses the relational assignment operator Can use WHERE or WHERE NOT to limit result INSERT uses assignment to set the relation to a union of the old version and the new tuple DELETE uses assignment to set the relation to the old version without the deleted tuple Relational operators are set operators: updates to individual tuples or attributes assign the entire relation during the operation

32. Interpreting Relvars Reminder: The heading of a relvar is a predicate, and the body of its instantiated relation is the set of true propositions The Closed World Assumption, a/k/a The Closed World Interpretation: If an otherwise valid tuple does not appear in the body of the relation, then the corresponding proposition is false

33. SQL Facilities – Rows SQL uses rows instead of tuples, with ordered columns Columns can be assigned values positionally If two rows are equal, this does not imply they are the same row Greater than and less than are legal row comparison operators SQL does not support row relational operators, such as row project

34. SQL Facilities – Table types SQL supports tables, not relations SQL table is not a relation of tuples, no, it is a bag of rows Bag is defined as an unordered set that allows duplicates Recap: in an SQL table, columns are ordered; rows are not

35. SQL Facilities – Table Values and Variables SQL uses “TABLE” ambiguously CREATE TABLE is used to establish a relvar, and the empty relation The initial INSERT establishes the relation, in the sense that it instantiates a table No support for table-valued columns No support for tables with zero columns Default is NULL

36. SQL Facilities – Table DROP and ALTER SQL supports DROP TABLE Can CASCADE or RESTRICT Table structures are ALTER’ed not updated Add column, change defaults or constraints

37. SQL Facilities – Structured Types CREATE TYPE POINT AS (X FLOAT,Y FLOAT) NOT FINAL; This can then be used as a type for columns of a table Types can be as simple or as complex as needed Structured types can be defined in terms of simpler types, and they can be named SQL support for objects