Bài giảng Database systems - Relational data model (3)

Domain Relational Calculus (1)  The formal specification of the domain calculus was proposed after the development of the QBE language and system.  Domain calculus differs from tuple calculus in the type of variables used in formulas: the variables range over single values from domains of attributes.  To form a relation of degree n for a query result, we must have n of these domain variables - one for each attribute.

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DATABASE SYSTEMS Nguyen Ngoc Thien An RELATIONAL DATA MODEL Spring 2014 Contents 2 Overview of Relational Data Model ER- & EER-to-Relational Mapping Relational Algebra Relational Algebra 3 Contents 4  Overview  Unary Relational Operations  Relational Algebra Operations from Set Theory  Binary Relational Operations  Additional Relational Operations  Brief Introduction to Relational Calculus Reading Suggestion: [1] Chapter 6 Overview (1) 5  A data model must include  Concepts for defining the database’s structure and constraints.  A set of operations to manipulate the database.  Relational Data Model  Structure and constraints  relation, attribute, tuple, integrity constraints.  Operations  Formal languages: the relational algebra and the relational calculus.  Practical language: SQL (Chapter 5). Overview (2) 6  Relational Algebra  A basic set of operations for the relational data model.  Specify retrieval requests (or queries) as relational algebra expressions.  Relational algebra expression: Sequence of relational algebra operations.  The result of an operation is a new relation, which may have been formed from one or more input relations.  This property makes the algebra “closed” (all objects in relational algebra are relations).  Importance:  A formal foundation for relational model operations.  A basis for implementing and optimizing queries in relational DBMSs.  Its concepts are incorporated into the SQL standard query language. Overview (3) 7  Relational Calculus  A higher-level declarative language for specifying relational queries.  Based on the branch of mathematical logic called predicate calculus.  A relational calculus expression creates a new relation.  Not specify how to retrieve the query result.  Only specify what information the result should contain.  Importance:  Having a firm basis in mathematical logic.  SQL has some of its foundations.  Two variations of relational calculus:  Tuple relational calculus: variables range over tuples.  Domain relational calculus: variables range over the domains (values) of attributes. Relational Algebra 8  Set operations from mathematical set theory:  UNION (∪), INTERSECTION (∩), DIFFERENCE (or MINUS,−)  CARTESIAN PRODUCT (×)  Operations developed specifically for relational databases:  Unary Relational Operations:  SELECT  symbol: 𝝈 (sigma)  PROJECT  symbol: 𝝅 (pi)  RENAME  symbol: 𝝆 (rho)  Binary Relational Operations:  JOIN (several variations of JOIN exist)  DIVISION  Additional Relational Operations:  OUTER JOINs, OUTER UNIONs  Aggregate functions (SUM, COUNT, AVG, MIN, MAX) 9 Fname Minit Lname SSN Bdate Address Sex Salary Super_SSN Dno Dname Dnumber Mgr_SSN Mgr_start_date Dnumber Dlocation PnumberPname Plocation Dnum Essn Pno Hours Essn Dependent_name BdateSex Relationship EMPLOYEE DEPARTMENT DEPT_LOCATIONS PROJECT WORKS_ON DEPENDENT 10 Contents 11  Overview  Unary Relational Operations  Relational Algebra Operations from Set Theory  Binary Relational Operations  Additional Relational Operations  Brief Introduction to Relational Calculus Unary Operations – SELECT (1) 12  The SELECT operation  Select a subset of the tuples from a relation based on a selection condition  filter/restrict.  Horizontal partition of the relation.  Denotation: 𝑆 = 𝜎(𝑅)  𝜎 (sigma): the SELECT operator.  𝑅: a relational algebra expression whose result is a relation.  : Boolean expression on attributes of R.  Result: tuples that make the condition true. Unary Operations – SELECT (2) 13  The SELECT operation  Properties:  𝐷𝑒𝑔𝑟𝑒𝑒(𝑆) = 𝐷𝑒𝑔𝑟𝑒𝑒(𝑅): S has the same attributes as R.  |𝑆| ≤ |𝑅|: The number of tuples of S is less than (or equal to) R.  SELECT is commutative: 𝜎 𝜎 𝑅 = 𝜎 𝜎 𝑅  Can combine a cascade (or sequence) of SELECT operations into a single SELECT operation: 𝜎 𝜎 𝜎 𝑅 = 𝜎 𝜎 𝜎 𝑅 = ⋯ = 𝜎𝐴𝑁𝐷𝐴𝑁𝐷(𝑅) Unary Operations – SELECT (3) 14  The Boolean expression  Made up of a number of clauses of the form: or  : one of the operators { =, , ≥, ≠, 𝑆𝑈𝐵𝑆𝑇𝑅𝐼𝑁𝐺_𝑂𝐹, } (dependent on domains).  Clauses can be connected by the standard Boolean operators 𝐴𝑁𝐷, 𝑂𝑅, 𝑁𝑂𝑇 to form a general selection condition. Unary Operations – SELECT (4) 15  E.g.:  Select the EMPLOYEE tuples whose department number is 4: 𝜎𝐷𝑛𝑜=4(𝐸𝑀𝑃𝐿𝑂𝑌𝐸𝐸)  Select the employee tuples whose salary is greater than $30,000: 𝜎𝑆𝑎𝑙𝑎𝑟𝑦>30,000(𝐸𝑀𝑃𝐿𝑂𝑌𝐸𝐸)  Select the tuples for all employees who either work in department 4 and make over $25,000 per year, or work in department 5 and make over $30,000: 𝜎 𝐷𝑛𝑜=4 𝑨𝑵𝑫 𝑆𝑎𝑙𝑎𝑟𝑦>25,000 𝑶𝑹 (𝐷𝑛𝑜=5 𝑨𝑵𝑫 𝑆𝑎𝑙𝑎𝑟𝑦>30,000)(𝐸𝑀𝑃𝐿𝑂𝑌𝐸𝐸) Unary Operations – PROJECT (1) 16  The PROJECT operation  Select certain columns from a relation and discard the other columns  project the relation over these attributes only.  Vertical partition of the relation.  Denotation: 𝑆 = 𝜋(𝑅)  𝜋 (pi): the PROJECT operator.  𝑅: a relational algebra expression whose result is a relation.  : the desired sublist of attributes from R.  Result: has only the attributes specified in < 𝑎𝑡𝑡𝑟𝑖𝑏𝑢𝑡𝑒 𝑙𝑖𝑠𝑡 > in the same order as they appear in the list. Unary Operations – PROJECT (2) 17  The PROJECT operation  Properties:  The degree of S is equal to the number of attributes in .  Duplicate elimination: The PROJECT operation removes any duplicate tuples  𝑆 ≤ 𝑅  If the list of attributes includes a superkey of R: 𝑆 = 𝑅 .  PROJECT is not commutative: 𝜋 𝜋 𝑅 = 𝜋(𝑅) as long as contains the attributes in ; otherwise, the left-hand side is an incorrect expression. Unary Operations – PROJECT (3) 18  E.g.:  List each employee’s first and last name and salary: 𝜋𝐿𝑛𝑎𝑚𝑒,𝐹𝑛𝑎𝑚𝑒,𝑆𝑎𝑙𝑎𝑟𝑦(𝐸𝑀𝑃𝐿𝑂𝑌𝐸𝐸)  List sex and salary of employees: 𝜋𝑆𝑒𝑥,𝑆𝑎𝑙𝑎𝑟𝑦(𝐸𝑀𝑃𝐿𝑂𝑌𝐸𝐸) Relational Algebra Expressions 19  Specify several relational algebra operations one after the other by one of two ways:  Write the operations as a single relational algebra expression by nesting the operations.  Apply one operation at a time and create intermediate result relations.  Must give names to the relations that hold the intermediate results. Unary Operations – RENAME (1) 20  The RENAME operation  Change the name of a relation or its attributes or both.  Useful when a query requires multiple operations.  Necessary in some cases (JOIN, UNION,).  Without renaming, the attribute names and their order in the resulting relation:  SELECT: the same as those in the original relation.  PROJECT: the same as those in the projection list. Unary Operations – RENAME (2) 21  The RENAME operation  Denotation:  Change the relation name only: 𝜌𝑆 𝑅  Change the attribute names only: 𝜌 𝐵1,𝐵2,,𝐵𝑛 (𝑅)  Change both the relation name and the attribute names: 𝜌𝑆 𝐵1,𝐵2,,𝐵𝑛 𝑅  𝜌 (rho): the RENAME operator.  𝑆: the new relation name.  𝐵1, 𝐵2, , 𝐵𝑛: the new attribute names. Unary Operations – RENAME (3) 22  E.g.:  A single relational algebra expression: 𝜋𝐹𝑛𝑎𝑚𝑒,𝐿𝑛𝑎𝑚𝑒,𝑆𝑎𝑙𝑎𝑟𝑦(𝜎𝐷𝑛𝑜=5 𝐸𝑀𝑃𝐿𝑂𝑌𝐸𝐸 )  Give a name to each intermediate relation: 𝑇𝐸𝑀𝑃 ← 𝜎𝐷𝑛𝑜=5 𝐸𝑀𝑃𝐿𝑂𝑌𝐸𝐸 𝑅 (𝐹𝑖𝑟𝑠𝑡_𝑛𝑎𝑚𝑒,𝐿𝑎𝑠𝑡_𝑛𝑎𝑚𝑒,𝑆𝑎𝑙𝑎𝑟𝑦) ← 𝜋𝐹𝑛𝑎𝑚𝑒,𝐿𝑛𝑎𝑚𝑒,𝑆𝑎𝑙𝑎𝑟𝑦 𝑇𝐸𝑀𝑃 Contents 23  Overview  Unary Relational Operations  Relational Algebra Operations from Set Theory  Binary Relational Operations  Additional Relational Operations  Brief Introduction to Relational Calculus Set Operations 24  Include: UNION, INTERSECTION, MINUS (also called SET DIFFERENCE or EXCEPT).  Merge the elements of two sets.  These are binary operations (applied to two sets of tuples).  Two relations on which a operator is applied must have the same type of tuples.  Type compatibility.  Type compatibility (Union compatibility): Two relations 𝑅 𝐴1, 𝐴2, , 𝐴𝑛 and 𝑆 𝐵1, 𝐵2, , 𝐵𝑛 are type compatible if they have:  The same degree 𝑛.  𝑑𝑜𝑚 𝐴𝑖 = 𝑑𝑜𝑚 𝐵𝑖 𝑓𝑜𝑟 1 ≤ 𝑖 ≤ 𝑛.  Convention: the resulting relation has the same attribute names as the first relation R. Set Operations – UNION 25  The UNION operation  Denotation: 𝑅 ∪ 𝑆  The result of this operation is a relation including all tuples that are either in R or in S or in both R and S.  Duplicate tuples are eliminated.  UNION is commutative: 𝑅 ∪ 𝑆 = 𝑆 ∪ 𝑅  UNION is associative: 𝑅 ∪ (𝑆 ∪ 𝑇) = (𝑅 ∪ 𝑆) ∪ 𝑇 Set Operations – INTERSECTION 26  The INTERSECTION operation  Denotation: 𝑅 ∩ 𝑆  The result of this operation is a relation including all tuples that are in both R and S.  INTERSECTION is commutative: 𝑅 ∩ 𝑆 = 𝑆 ∩ 𝑅  INTERSECTION is associative: 𝑅 ∩ (𝑆 ∩ 𝑇) = (𝑅 ∩ 𝑆) ∩ 𝑇  INTERSECTION can be expressed in terms of union and set difference: 𝑅 ∩ 𝑆 = 𝑅 ∪ 𝑆 − 𝑅 − 𝑆 − (𝑆 − 𝑅) Set Operations – MINUS 27  The MINUS (or SET DIFFERENCE) operation  Denotation: 𝑅 − 𝑆  The result of this operation is a relation including all tuples that are in R but not in S.  MINUS is not commutative: 𝑅 − 𝑆 ≠ 𝑆 − 𝑅 Set Operations – Example (1) 28  E.g.:  Retrieve the Ssn of all employees who either work in department 5 or directly supervise an employee who works in department 5: 𝐷𝐸𝑃5_𝐸𝑀𝑃𝑆 ← 𝜎𝐷𝑛𝑜=5 𝐸𝑀𝑃𝐿𝑂𝑌𝐸𝐸 𝑅𝐸𝑆𝑈𝐿𝑇1 ← 𝜋𝑆𝑠𝑛(𝐷𝐸𝑃5_𝐸𝑀𝑃𝑆) 𝑅𝐸𝑆𝑈𝐿𝑇2 (𝑆𝑠𝑛) ← 𝜋𝑆𝑢𝑝𝑒𝑟_𝑆𝑠𝑛(𝐷𝐸𝑃5_𝐸𝑀𝑃𝑆) 𝑅𝐸𝑆𝑈𝐿𝑇 ← 𝑅𝐸𝑆𝑈𝐿𝑇1 ∪ 𝑅𝐸𝑆𝑈𝐿𝑇2 Set Operations – Example (2) 29 Set Operations - CARTESIAN PRODUCT 30  The CARTESIAN PRODUCT operation  Also known as CROSS PRODUCT or CROSS JOIN.  A binary set operation.  Two operand relations do NOT have to be type compatible.  Produce a new element by combining every member (tuple) from one relation (set) with every member from the other relation.  Denotation: 𝑅 × 𝑆 (with 𝑅 𝐴1, 𝐴2, , 𝐴𝑛 , 𝑆(𝐵1, 𝐵2, , 𝐵𝑚))  Result: 𝑄(𝐴1, 𝐴2, , 𝐴𝑛, 𝐵1, 𝐵2, , 𝐵𝑚) , in that order.  Properties:  𝐷𝑒𝑔𝑟𝑒𝑒 𝑄 = 𝑛 + 𝑚  𝑅 = 𝑛𝑅, 𝑆 = 𝑛𝑆  𝑅 × 𝑆 = 𝑛𝑅 ∗ 𝑛𝑆  Useful when followed by a selection that matches values of attributes. Contents 31  Overview  Unary Relational Operations  Relational Algebra Operations from Set Theory  Binary Relational Operations  Additional Relational Operations  Brief Introduction to Relational Calculus Binary Operations – JOIN (1) 32  The JOIN operation  Combine related tuples from two relations into single “longer” tuples.  Denotation: 𝑅 ⋈ 𝑆  𝑅 𝐴1, 𝐴2, , 𝐴𝑛 , 𝑆(𝐵1, 𝐵2, , 𝐵𝑚) : relations that result from general relational algebra expressions.  = AND AND AND : The join condition is specified on attributes from R and S and is evaluated for each combination of tuples.  = 𝐴𝑖 𝜃 𝐵𝑗  𝐴𝑖: an attriute of R 𝐵𝑗: an attribute of S  𝐴𝑖 𝑎𝑛𝑑 𝐵𝑗 : have the same domain.  𝜃: one of the comparison operators {=, , ≥, ≠}. Binary Operations – JOIN (2) 33  The JOIN operation  Result:  Only combinations of tuples satisfying the join condition appear in Q as a single combined tuple 𝑄(𝐴1, 𝐴2, , 𝐴𝑛, 𝐵1, 𝐵2, , 𝐵𝑚) , in that order.  Tuples whose join attributes are NULL or for which the join condition is FALSE do not appear in the result.  Properties:  𝐷𝑒𝑔𝑟𝑒𝑒 𝑄 = 𝑛 + 𝑚  𝑅 = 𝑛𝑅, 𝑆 = 𝑛𝑆  0 ≤ 𝑄 ≤ 𝑛𝑅 ∗ 𝑛𝑆  The expected size of the join result divided by the maximum size 𝑛𝑅 ∗ 𝑛𝑆 leads to a ratio called join selectivity.  Can be specified as a CARTESIAN PRODUCT operation followed by a SELECT operation.  A JOIN operation with a general join condition is called a THETA JOIN. Binary Operations – JOIN (3) 34  E.g.:  Retrieve the name of the manager of each department: 𝐷𝐸𝑃𝑇_𝑀𝐺𝑅 ← 𝐷𝐸𝑃𝐴𝑅𝑇𝑀𝐸𝑁𝑇 ⋈𝑀𝑔𝑟_𝑆𝑠𝑛=𝑆𝑠𝑛 𝐸𝑀𝑃𝐿𝑂𝑌𝐸𝐸 𝑅𝐸𝑆𝑈𝐿𝑇 ← 𝜋𝐷𝑛𝑎𝑚𝑒,𝐿𝑛𝑎𝑚𝑒,𝐹𝑛𝑎𝑚𝑒(𝐷𝐸𝑃𝑇_𝑀𝐺𝑅) Binary Operations – JOIN (4) 35  The EQUIJOIN operation  A join where the only comparison operator used is “=“.  The result of an EQUIJOIN always has one or more pairs of attributes (whose names need NOT to be identical) that have identical values in every tuple. Binary Operations – JOIN (5) 36  The NATURAL JOIN operation  Created to get rid of the second (superfluous) attribute in an EQUIJOIN condition.  The standard definition of NATURAL JOIN requires that the two join attributes, or each pair of corresponding join attributes, have to have the same name in both relations.  If not, a RENAME operation is applied first.  Denotation: 𝑅 ∗ 𝑆 Binary Operations – JOIN (6) 37 Binary Operations – DIVISION (1) 38  The DIVISION operation  Denotation: 𝑅 ÷ 𝑆  Applied to two relations 𝑅(𝑍) ÷ 𝑆(𝑋) where 𝑍 = 𝑋 ∪ 𝑌 (Y is the set of attributes of R that are not attributes of S).  Result: a relation 𝑇(𝑌) that includes a tuple t if tuples 𝑡𝑅 appear in 𝑅 with 𝑡𝑅 𝑌 = 𝑡 , and with 𝑡𝑅 𝑋 = 𝑡𝑆 for every tuple 𝑡𝑆 in S, i.e., for a tuple t to appear in the result T of the DIVISION, the values in t must appear in R in combination with every tuple in S. Binary Operations – DIVISION (2) 39 Complete Set of Relational Operations 40  {𝜎, 𝜋,∪, 𝜌, −,×} : complete set of relational algebra operations.  Any of the other original relational algebra operations can be expressed as a sequence of operations from this set.  E.g.:  𝑅 ∩ 𝑆 ≡ 𝑅 ∪ 𝑆 − ( 𝑅 − 𝑆 ∪ 𝑆 − 𝑅 )  𝑅 ⋈ 𝑆 ≡ 𝜎(𝑅 × 𝑆) Notation for Query Trees 41  Query Tree  Represents the input relations of query as leaf nodes of the tree.  Represents the relational algebra operations as internal nodes. Contents 42  Overview  Unary Relational Operations  Relational Algebra Operations from Set Theory  Binary Relational Operations  Additional Relational Operations  Brief Introduction to Relational Calculus Additional Relational Operations (1) 43  Generalized Projection  Denotation: 𝑆 = 𝜋𝐹1,𝐹2,,𝐹𝑛(𝑅)  𝐹1, 𝐹2, , 𝐹𝑛: functions over the attributes in relation R and may involve arithmetic operations and constant values  E.g.: 𝑅𝐸𝑃𝑂𝑅𝑇 ← 𝜌 𝑆𝑠𝑛,𝑁𝑒𝑡𝑆𝑎𝑙𝑎𝑟𝑦,𝐵𝑜𝑛𝑢𝑠,𝑇𝑎𝑥 (𝜋𝑆𝑠𝑛,𝑆𝑎𝑙𝑎𝑟𝑦 −𝐷𝑒𝑑𝑢𝑐𝑡𝑖𝑜𝑛, 2000∗𝑌𝑒𝑎𝑟𝑆𝑒𝑟𝑣𝑖𝑐𝑒, 0.25∗𝑆𝑎𝑙𝑎𝑟𝑦 𝐸𝑀𝑃𝐿𝑂𝑌𝐸𝐸 ) Additional Relational Operations (2) 44  Aggregate Functions  Specify mathematical aggregate functions on collections of values from the database.  E.g.: retrieving the average or total salary of all employees or the total number of employee tuples.  Common functions applied to collections of numeric values include SUM, AVERAGE, MAXIMUM, and MINIMUM.  COUNT function: used for counting tuples or values.  Grouping  Group the tuples in a relation by the value of some of their attributes and apply an aggregate function independently to each group.  E.g.: Group EMPLOYEE tuples by Dno and list each Dno value along with the average salary of employees within the department. Additional Relational Operations (3) 45  Denotation: ℱ(𝑅)  : list of attributes of the relation specified in R.  : list of ( < 𝑎𝑡𝑡𝑟𝑖𝑏𝑢𝑡𝑒 >) pairs.  : one of the allowed functions (SUM, AVERAGE , MAXIMUM, MINIMUM, COUNT).  : an attribute of the relation specified by R.  Resulting relation has the grouping attributes plus one attribute for each element in the function list.  If no grouping attributes are specified, the functions are applied to all the tuples in R. Additional Relational Operations (4) 46  Use of the Functional operator ℱ  ℱ𝑀𝐴𝑋 𝑆𝑎𝑙𝑎𝑟𝑦 𝐸𝑀𝑃𝐿𝑂𝑌𝐸𝐸 retrieves the maximum salary value from the Employee relation.  ℱ𝑀𝐼𝑁 𝑆𝑎𝑙𝑎𝑟𝑦 𝐸𝑀𝑃𝐿𝑂𝑌𝐸𝐸 retrieves the minimum Salary value from the Employee relation.  ℱ𝑆𝑈𝑀 𝑆𝑎𝑙𝑎𝑟𝑦 𝐸𝑀𝑃𝐿𝑂𝑌𝐸𝐸 retrieves the sum of the Salary from the Employee relation.  𝐷𝑛𝑜ℱ𝐶𝑂𝑈𝑁𝑇 𝑆𝑠𝑛,𝐴𝑉𝐸𝑅𝐴𝐺𝐸 𝑆𝑎𝑙𝑎𝑟𝑦 𝐸𝑀𝑃𝐿𝑂𝑌𝐸𝐸 groups employees by DNO (department number) and computes the count of employees and average salary per department.  Note: Duplicates are NOT eliminated when an aggregate function is applied. Additional Relational Operations (5) 47  E.g.:  𝐷𝑛𝑜ℱ𝐶𝑂𝑈𝑁𝑇 𝑆𝑠𝑛,𝐴𝑉𝐸𝑅𝐴𝐺𝐸 𝑆𝑎𝑙𝑎𝑟𝑦 𝐸𝑀𝑃𝐿𝑂𝑌𝐸𝐸  ℱ𝐶𝑂𝑈𝑁𝑇 𝑆𝑠𝑛,𝐴𝑉𝐸𝑅𝐴𝐺𝐸 𝑆𝑎𝑙𝑎𝑟𝑦 𝐸𝑀𝑃𝐿𝑂𝑌𝐸𝐸 Additional Relational Operations (6) 48  Recursive Closure Operations  In general, recursive closure cannot be specified in the basic original relational algebra. This operation is applied to a recursive relationship.  An example of a recursive operation is to retrieve all SUPERVISEES of an EMPLOYEE e at all levels.  We cannot, in general, specify a query such as “retrieve the supervisees of ‘James Borg’ at all levels” without utilizing a looping mechanism unless we know the maximum number of levels.  The SQL3 standard includes syntax for recursive closure.  Details: Read [1] 6.4.3 Additional Relational Operations (7) 49  The INNER JOIN operation  Tuples with no match are eliminated.  Tuples without a matching (or related) tuple are eliminated from the join result.  Tuples with NULL in the join attributes are also eliminated.  Include: THETA JOIN, EQUIJOIN, NATURAL JOIN.  The OUTER JOIN operation  Keep all the tuples in R, or all those in S, or all those in both relations in the result of the JOIN, regardless of whether or not they have matching tuples in the other relation.  Include: LEFT OUTER JOIN ⊐⋈, RIGHT OUTER JOIN, FULL OUTER JOIN.  The Outer Union operation: Read [1] 6.4.5 Additional Relational Operations (8) 50  The LEFT OUTER JOIN operation  Denotation: 𝑅 ⋈ 𝑆  Keep every tuple in the first, or left, relation R. If no matching tuple is found in S, then the attributes of S in the join result are filled or “padded” with NULL values.  The RIGHT OUTER JOIN operation  Denotation: 𝑅 ⋈ 𝑆  Keep every tuple in the second, or right, relation S. If no matching tuple is found in R, then the attributes of R in the join result are filled or “padded” with NULL values.  The FULL OUTER JOIN operation  Denotation: 𝑅 ⋈ 𝑆  Keep all tuples in both the left and the right relations when no matching tuples are found, padding them with NULL values as needed. Additional Relational Operations (9) 51  E.g.: List all employee names and also the name of the departments they manage if they happen to manage a department (if they do not manage one, we can indicate it with a NULL value). Homework !!! 52  Reading: [1] 6.5 (Examples of Queries in Relational Algebra) Contents 53  Overview  Unary Relational Operations  Relational Algebra Operations from Set Theory  Binary Relational Operations  Additional Relational Operations  Brief Introduction to Relational Calculus Relational Calculus 54  A relational calculus expression creates a new relation, which is specified in terms of variables that range over rows (in tuple calculus) or over columns of the stored relations (in domain calculus).  In a calculus expression, there is no order of operations to specify how to retrieve the query result.  A calculus expression specifies only what information the result should contain.  Relational calculus is considered to be a nonprocedural language.  Relational algebra can be considered as a procedural way of stating a query. Tuple Relational Calculus (1) 55  Based on specifying a number of tuple variables. Each tuple variable usually ranges over a particular database relation, meaning that the variable may take as its value any individual tuple from that relation.  A simple tuple relational calculus query is of the form {𝑡 | 𝐶𝑂𝑁𝐷(𝑡)} where t is a tuple variable and 𝐶𝑂𝑁𝐷 (𝑡) is a conditional expression involving t.  E.g.: To find the first and last names of all employees whose salary is above $50,000: {𝑡. 𝐹𝑁𝐴𝑀𝐸, 𝑡. 𝐿𝑁𝐴𝑀𝐸 | 𝐸𝑀𝑃𝐿𝑂𝑌𝐸𝐸(𝑡) 𝐴𝑁𝐷 𝑡. 𝑆𝐴𝐿𝐴𝑅𝑌 > 50000} Tuple Relational Calculus (2) 56  Quantifiers can appear in formulas:  The universal quantifier: ∀  The existential quantifier: ∃  Informally, a tuple variable t is bound if it is quantified, meaning that it appears in an (∃𝑡) or (∀𝑡) clause; otherwise, it is free.  Details: Read [1] 6.6 Tuple Relational Calculus (3) 57  E.g.:  Retrieve the name and address of all employees who work for the ‘Research’ department. 𝑡. 𝐹𝑛𝑎𝑚𝑒, 𝑡. 𝐿𝑛𝑎𝑚𝑒, 𝑡. 𝐴𝑑𝑑𝑟𝑒𝑠𝑠 𝐸𝑀𝑃𝐿𝑂𝑌𝐸𝐸 𝑡 𝑨𝑵𝑫 (∃ 𝑑) (𝐷𝐸𝑃𝐴𝑅𝑇𝑀𝐸𝑁𝑇 𝑑 𝑨𝑵𝑫 𝑑. 𝐷𝑛𝑎𝑚𝑒 = ‘𝑅𝑒𝑠𝑒𝑎𝑟𝑐ℎ’ 𝑨𝑵𝑫 𝑑. 𝐷𝑛𝑢𝑚𝑏𝑒𝑟 = 𝑡. 𝐷𝑛𝑜) }  Find the names of employees who work on some projects controlled by department number 5. {𝑒. 𝐿𝑛𝑎𝑚𝑒, 𝑒. 𝐹𝑛𝑎𝑚𝑒|𝐸𝑀𝑃𝐿𝑂𝑌𝐸𝐸(𝑒) 𝑨𝑵𝑫((∃ 𝑥)(∃ 𝑤)(𝑃𝑅𝑂𝐽𝐸𝐶𝑇 𝑥 𝑨𝑵𝑫 𝑊𝑂𝑅𝐾𝑆_𝑂𝑁 𝑤 𝑨𝑵𝑫 𝑥. 𝐷𝑛𝑢𝑚𝑏𝑒𝑟 = 5 𝑨𝑵𝑫 𝑤. 𝐸𝑆𝑆𝑁 = 𝑒. 𝑆𝑆𝑁 𝑨𝑵𝑫 𝑥. 𝑃𝑁𝑈𝑀𝐵𝐸𝑅 = 𝑤. 𝑃𝑁𝑂))} Domain Relational Calculus (1) 58  The formal specification of the domain calculus was proposed after the development of the QBE language and system.  Domain calculus differs from tuple calculus in the type of variables used in formulas: the variables range over single values from domains of attributes.  To form a relation of degree n for a query result, we must have n of these domain variables - one for each attribute.  An expression of the domain calculus is of the form: 𝑥1, 𝑥2, , 𝑥𝑛 𝐶𝑂𝑁𝐷 𝑥1, 𝑥2, , 𝑥𝑛, 𝑥𝑛+1, 𝑥𝑛+2, , 𝑥𝑛+𝑚 } Where 𝑥1, 𝑥2, , 𝑥𝑛 , 𝑥𝑛+1, 𝑥𝑛+2, , 𝑥𝑛+𝑚 are domain variables that range over domains (of attributes) and 𝐶𝑂𝑁𝐷 is a condition or formula of the domain relational calculus.  Details: Read [1] 6.7 Domain Relational Calculus (2) 59  E.g.: List the birth date and address of the employee whose name is ‘John B. Smith’. {𝑢, 𝑣| ∃𝑞 ∃𝑟 ∃𝑠 ∃𝑡 ∃𝑤 ∃𝑥 ∃𝑦 ∃𝑧 𝐸𝑀𝑃𝐿𝑂𝑌𝐸𝐸 𝑞𝑟𝑠𝑡𝑢𝑣𝑤𝑥𝑦𝑧 𝐴𝑁𝐷 𝑞 =′ 𝐽𝑜ℎ𝑛′ 𝐴𝑁𝐷 𝑟 =′ 𝐵′𝐴𝑁𝐷𝑠 = ′𝑆𝑚𝑖𝑡ℎ′ } Q & A 60

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