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Relational algebra.pptx
1.
©Silberschatz, Korth and
Sudarshan3.1Database System Concepts Chapter 3: Relational Model Structure of Relational Databases Relational Algebra Tuple Relational Calculus Domain Relational Calculus Extended Relational-Algebra-Operations Modification of the Database Views
2.
©Silberschatz, Korth and
Sudarshan3.2Database System Concepts Example of a Relation
3.
©Silberschatz, Korth and
Sudarshan3.3Database System Concepts Relation Instance The current values (relation instance) of a relation are specified by a table An element t of r is a tuple, represented by a row in a table Jones Smith Curry Lindsay customer-name Main North North Park customer-street Harrison Rye Rye Pittsfield customer-city customer attributes (or columns) tuples (or rows)
4.
©Silberschatz, Korth and
Sudarshan3.4Database System Concepts Query Language A query language is a language in which a user requests information from the database. These languages are usually on a level higher than that of a standard programming language. Query languages can be categorized as either procedural or nonprocedural. In a procedural language, the user instructs the system to perform a sequence of operations on the database to compute the desired result. In a nonprocedural language, the user describes the desired information without giving a specific procedure for obtaining that information. The relational algebra is procedural. the tuple relational calculus and domain relational calculus are nonprocedural.
5.
©Silberschatz, Korth and
Sudarshan3.5Database System Concepts RELATIONAL ALGEBRA The relational algebra is a procedural query language. It consists of a set of operations that take one or two relations (tables) as input and produce a new relation, on the request of the user to retrieve the specific information, as the output. The relational algebra contains the following operations
6.
©Silberschatz, Korth and
Sudarshan3.6Database System Concepts Relational Algebra Procedural language Six basic operators select project union set difference Cartesian product rename The operators take one or more relations as inputs and give a new relation as a result.
7.
©Silberschatz, Korth and
Sudarshan3.7Database System Concepts
8.
©Silberschatz, Korth and
Sudarshan3.8Database System Concepts Unary Relational Operations SELECT (symbol: σ) PROJECT (symbol: π) RENAME (symbol: ρ) Relational Algebra Operations From Set Theory UNION (υ) INTERSECTION ( ), DIFFERENCE (-) CARTESIAN PRODUCT ( x ) Binary Relational Operations JOIN DIVISION
9.
©Silberschatz, Korth and
Sudarshan3.9Database System Concepts SELECTION (σ) • The SELECT operation is used for selecting a subset of the tuples according to a given selection condition. Sigma(σ)Symbol denotes it. • Ex:- find all employees born after 1st Jan 1950: • dob σ '01/JAN/1950'(employee)
10.
©Silberschatz, Korth and
Sudarshan3.10Database System Concepts σp(r) σ is the predicate r stands for relation which is the name of the table p is prepositional logic
11.
©Silberschatz, Korth and
Sudarshan3.11Database System Concepts Example 1 σ topic = "Database" (Tutorials) Output - Selects tuples from Tutorials where topic = 'Database'. Example 2 σ topic = "Database" and author = "guru99"( Tutorials) Output - Selects tuples from Tutorials where the topic is 'Database' and 'author' is guru99. Example 3 σ sales > 50000 (Customers) Output - Selects tuples from Customers where sales is greater than 50000
12.
©Silberschatz, Korth and
Sudarshan3.12Database System Concepts sSalary > 40000 (Employee) SSN Name Salary 1234545 John 200000 5423341 Smith 600000 4352342 Fred 500000 SSN Name Salary 5423341 Smith 600000 4352342 Fred 500000
13.
©Silberschatz, Korth and
Sudarshan3.13Database System Concepts σsubject = "database"(Books) Output − Selects tuples from books where subject is 'database'. σsubject = "database" and price = "450"(Books) Output − Selects tuples from books where subject is 'database' and 'price' is 450. σsubject = "database" and price = "450" or year > "2010"(Books) Output − Selects tuples from books where subject is 'database' and 'price' is 450 or those books published after 2010.
14.
©Silberschatz, Korth and
Sudarshan3.14Database System Concepts PROJECTION(∏ )Pi • ∏ (pi) symbol used to choose attributes from a relation. This helps to extract the values of specified attributes to eliminates duplicate values. (pi) symbol is used to choose attributes from a relation.
15.
©Silberschatz, Korth and
Sudarshan3.15Database System Concepts SELECTION & PROJECTION Example Id Person Name Address Hobby 1123 1123 5556 9876 John John Mary Bart 123 Main 123 Main 7 Lake Dr 5 Pine St stamps coins hiking stamps Id Hobby 1123 Name Address John 123 Main stamps 9876 Bart 5 Pine St stamps σ Hobby=‘stamps’(Person) ∏Name,Hobby(Person) Name Hobby John John Mary Bart stamps coins Hiking stamps
16.
©Silberschatz, Korth and
Sudarshan3.16Database System Concepts P Name,Salary (Employee) SSN Name Salary 1234545 John 200000 5423341 John 600000 4352342 John 200000 Name Salary John 20000 John 60000
17.
©Silberschatz, Korth and
Sudarshan3.17Database System Concepts Project Operation – Example Relation r: A B C 10 20 30 40 1 1 1 2 A C 1 1 1 2 = A C 1 1 2 A,C (r)
18.
©Silberschatz, Korth and
Sudarshan3.18Database System Concepts CustomerID CustomerName Status 1 Google Active 2 Amazon Active 3 Apple Inactive 4 Alibaba Active Here, the projection of CustomerName and status will give Π CustomerName, Status (Customers) CustomerName Status Google Active Amazon Active Apple Inactive Alibaba Active
19.
©Silberschatz, Korth and
Sudarshan3.19Database System Concepts The results of relational algebra are also relations but without any name. The rename operation allows us to rename the output relation. 'rename' operation is denoted with small Greek letter rho ρ. Notation − ρ x (E) Where the result of expression E is saved with name of x 'rename'
20.
©Silberschatz, Korth and
Sudarshan3.20Database System Concepts UNION UNION is symbolized by ∪ symbol. It includes all tuples that are in tables A or in B. It also eliminates duplicate tuples. So, set A UNION set B would be expressed as: The result <- A ∪ B For a union operation to be valid, the following conditions must hold - R and S must be the same number of attributes. Attribute domains need to be compatible. Duplicate tuples should be automatically removed.
21.
©Silberschatz, Korth and
Sudarshan3.21Database System Concepts Union Operation – Example Relations r, s: r s: A B 1 2 1 A B 2 3 r s A B 1 2 1 3
22.
©Silberschatz, Korth and
Sudarshan3.22Database System Concepts TABLE A Table B column 1 column 2 column 1 column 2 1 1 1 1 1 2 1 3 A ∪ B gives Table A ∪ B column 1 column 2 1 1 1 2 1 3
23.
©Silberschatz, Korth and
Sudarshan3.23Database System Concepts Set Difference (-) - Symbol denotes it. The result of A - B, is a relation which includes all tuples that are in A but not in B. The attribute name of A has to match with the attribute name in B.
24.
©Silberschatz, Korth and
Sudarshan3.24Database System Concepts Set Difference Operation – Example Relations r, s: r – s: A B 1 2 1 A B 2 3 r s A B 1 1
25.
©Silberschatz, Korth and
Sudarshan3.25Database System Concepts What about Intersection ? An intersection is defined by the symbol ∩ A ∩ B Defines a relation consisting of a set of all tuple that are in both A and B. However, A and B must be union-compatible.
26.
©Silberschatz, Korth and
Sudarshan3.26Database System Concepts Table A ∩ B TABLE A Table B column 1 column 2 column 1 column 2 1 1 1 1 1 2 1 3 column 1 column 2 1 1
27.
©Silberschatz, Korth and
Sudarshan3.27Database System Concepts Cartesian-Product Operation artesian Product in DBMS is an operation used to merge columns from two relations. Notation − r Χ s Where r and s are relations and their output will be defined as − σauthor = 'tutorialspoint'(Books Χ Articles) Output − Yields a relation, which shows all the books and articles written by tutorialspoint.
28.
©Silberschatz, Korth and
Sudarshan3.28Database System Concepts Cartesian-Product Operation-Example Relations r, s: r x s: A B 1 2 A B 1 1 1 1 2 2 2 2 C D 10 10 20 10 10 10 20 10 E a a b b a a b b C D 10 10 20 10 E a a b br s
29.
©Silberschatz, Korth and
Sudarshan3.29Database System Concepts
30.
©Silberschatz, Korth and
Sudarshan3.30Database System Concepts Join Operations
31.
©Silberschatz, Korth and
Sudarshan3.31Database System Concepts
32.
©Silberschatz, Korth and
Sudarshan3.32Database System Concepts
33.
©Silberschatz, Korth and
Sudarshan3.33Database System Concepts
34.
©Silberschatz, Korth and
Sudarshan3.34Database System Concepts
35.
©Silberschatz, Korth and
Sudarshan3.35Database System Concepts
36.
©Silberschatz, Korth and
Sudarshan3.36Database System Concepts NATURAL JOIN (⋈) Natural join can only be performed if there is a common attribute (column) between the relations. The name and type of the attribute must be same.
37.
©Silberschatz, Korth and
Sudarshan3.37Database System Concepts
38.
©Silberschatz, Korth and
Sudarshan3.38Database System Concepts
39.
©Silberschatz, Korth and
Sudarshan3.39Database System Concepts Example: Find the employee names and city who have salary details. semp_name, salary, city ( employee ⋈ employee_works ) The join operation selects all employees with salary details, from where we can easily project the employee names, cities and salaries. Natural Join operation results in some loss of information
40.
©Silberschatz, Korth and
Sudarshan3.40Database System Concepts
41.
©Silberschatz, Korth and
Sudarshan3.41Database System Concepts
42.
©Silberschatz, Korth and
Sudarshan3.42Database System Concepts Right Outer Join:
43.
©Silberschatz, Korth and
Sudarshan3.43Database System Concepts
44.
©Silberschatz, Korth and
Sudarshan3.44Database System Concepts Outer Join – Example Relation loan Relation borrower customer-name loan-number Jones Smith Hayes L-170 L-230 L-155 3000 4000 1700 loan-number amount L-170 L-230 L-260 branch-name Downtown Redwood Perryridge
45.
©Silberschatz, Korth and
Sudarshan3.45Database System Concepts Outer Join – Example Inner Join loan Borrower loan-number amount L-170 L-230 3000 4000 customer-name Jones Smith branch-name Downtown Redwood Jones Smith null loan-number amount L-170 L-230 L-260 3000 4000 1700 customer-namebranch-name Downtown Redwood Perryridge Left Outer Join loan Borrower
46.
©Silberschatz, Korth and
Sudarshan3.46Database System Concepts Outer Join – Example Right Outer Join loan borrower loan borrower Full Outer Join loan-number amount L-170 L-230 L-155 3000 4000 null customer-name Jones Smith Hayes branch-name Downtown Redwood null loan-number amount L-170 L-230 L-260 L-155 3000 4000 1700 null customer-name Jones Smith null Hayes branch-name Downtown Redwood Perryridge null
47.
©Silberschatz, Korth and
Sudarshan3.47Database System Concepts
48.
©Silberschatz, Korth and
Sudarshan3.48Database System Concepts
49.
©Silberschatz, Korth and
Sudarshan3.49Database System Concepts
50.
©Silberschatz, Korth and
Sudarshan3.50Database System Concepts Banking Example branch (branch-name, branch-city, assets) customer (customer-name, customer-street, customer-only) account (account-number, branch-name, balance) loan (loan-number, branch-name, amount) depositor (customer-name, account-number) borrower (customer-name, loan-number)
51.
©Silberschatz, Korth and
Sudarshan3.51Database System Concepts Example Queries Find all loans of over $1200 Find the loan number for each loan of an amount greater than $1200 samount > 1200 (loan) loan-number (samount > 1200 (loan))
52.
©Silberschatz, Korth and
Sudarshan3.52Database System Concepts Example Queries Find the names of all customers who have a loan, an account, or both, from the bank Find the names of all customers who have a loan and an account at bank. customer-name (borrower) customer-name (depositor) customer-name (borrower) customer-name (depositor)
53.
©Silberschatz, Korth and
Sudarshan3.53Database System Concepts 53 Examples Sailors (sid, name, rating, age) bid name color 101 Interlake blue 102 Interlake red 103 Clipper green 104 Marine red Boats (bid, name, color) sid bid day 1 101 10/10/12 1 102 10/10/12 1 101 10/7/12 2 102 11/9/12 2 102 7/11/12 3 101 7/11/12 3 102 7/8/12 4 103 19/9/12 Reserves (sid, bid, day) List names of boats. List ratings and ages sailors. List names of sailors who are over 21 years old. List names of red boats. πname ( Boats) πrating, age (Sailors) πname ( σage>21 (Sailors))πname ( σcolor=red (Boats)) sid name rating age 1 Dustin 7 45 2 Rusty 10 35 3 Horatio 5 35 4 Zorba 8 18 5 Julius 25
54.
©Silberschatz, Korth and
Sudarshan3.54Database System Concepts 54 Examples Sailors (sid, name, rating, age) bid name color 101 Interlake blue 102 Interlake red 103 Clipper green 104 Marine red Boats (bid, name, color) sid bid day 1 101 10/10/12 1 102 10/10/12 1 101 10/7/12 2 102 11/9/12 2 102 7/11/12 3 101 7/11/12 3 102 7/8/12 4 103 19/9/12 Reserves (sid, bid, day) List ids of boats named Interlake List ids of boats reserved on 10/10/12 sid name rating age 1 Dustin 7 45 2 Rusty 10 35 3 Horatio 5 35 4 Zorba 8 18 5 Julius 25
55.
©Silberschatz, Korth and
Sudarshan3.55Database System Concepts 55 Examples (solution) Sailors (sid, name,rating, age) Boats (bid, name, color) Reserves (sid, bid,day)bid name color 101 Interlake blue 102 Interlake red 103 Clipper green 104 Marine red sid bid day 1 101 10/10/12 1 102 10/10/12 1 101 10/7/12 2 102 11/9/12 2 102 7/11/12 3 101 7/11/12 3 102 7/8/12 4 103 19/9/12List ids of boats named Interlake bid (snameInterlake(Boats)) List ids of boats reserved on 10/10/12 bid (sday10/10/12 (Reserves)) sid name rating age 1 Dustin 7 45 2 Rusty 10 35 3 Horatio 5 35 4 Zorba 8 18 5 Julius 25
56.
©Silberschatz, Korth and
Sudarshan3.56Database System Concepts 56 Examples Sailors (sid, name, rating, age) bid name color 101 Interlake blue 102 Interlake red 103 Clipper green 104 Marine red Boats (bid, name, color) sid bid day 1 101 10/10/12 1 102 10/10/12 1 101 10/7/12 2 102 11/9/12 2 102 7/11/12 3 101 7/11/12 3 102 7/8/12 4 103 19/9/12 Reserves (sid, bid, day) List ids of sailors who reserved boat 102 πsid ( σbid=102 Reserves) List names of sailors who reserved boat 102 πname (Sailors ⨝ ( σbid=102 Reserves)) πname (σbid=102 (Sailors ⨝ Reserves)) both are correct! which is better? sid name rating age 1 Dustin 7 45 2 Rusty 10 35 3 Horatio 5 35 4 Zorba 8 18 5 Julius 25
57.
©Silberschatz, Korth and
Sudarshan3.57Database System Concepts Example Queries Find the names of all customers having a loan at the Perryridge branch {t | s borrower( t[customer-name] = s[customer-name] u loan(u[branch-name] = “Perryridge” u[loan-number] = s[loan-number])) not v depositor (v[customer-name] = t[customer-name]) } Find the names of all customers who have a loan at the Perryridge branch, but no account at any branch of the bank {t | s borrower(t[customer-name] = s[customer-name] u loan(u[branch-name] = “Perryridge” u[loan-number] = s[loan-number]))}
58.
©Silberschatz, Korth and
Sudarshan3.58Database System Concepts Example Queries Find the names of all customers having a loan from the Perryridge branch, and the cities they live in {t | s loan(s[branch-name] = “Perryridge” u borrower (u[loan-number] = s[loan-number] t [customer-name] = u[customer-name]) v customer (u[customer-name] = v[customer-name] t[customer-city] = v[customer-city])))}
59.
©Silberschatz, Korth and
Sudarshan3.59Database System Concepts Example Queries Find the names of all customers who have an account at all branches located in Brooklyn: {t | c customer (t[customer.name] = c[customer-name]) s branch(s[branch-city] = “Brooklyn” u account ( s[branch-name] = u[branch-name] s depositor ( t[customer-name] = s[customer-name] s[account-number] = u[account-number] )) )}
60.
©Silberschatz, Korth and
Sudarshan3.60Database System Concepts Safety of Expressions It is possible to write tuple calculus expressions that generate infinite relations. For example, {t | t r} results in an infinite relation if the domain of any attribute of relation r is infinite To guard against the problem, we restrict the set of allowable expressions to safe expressions. An expression {t | P(t)} in the tuple relational calculus is safe if every component of t appears in one of the relations, tuples, or constants that appear in P NOTE: this is more than just a syntax condition. E.g. { t | t[A]=5 true } is not safe --- it defines an infinite set with attribute values that do not appear in any relation or tuples or constants in P.
61.
©Silberschatz, Korth and
Sudarshan3.61Database System Concepts Domain Relational Calculus A nonprocedural query language equivalent in power to the tuple relational calculus Each query is an expression of the form: { x1, x2, …, xn | P(x1, x2, …, xn)} x1, x2, …, xn represent domain variables P represents a formula similar to that of the predicate calculus
62.
©Silberschatz, Korth and
Sudarshan3.62Database System Concepts Example Queries Find the loan-number, branch-name, and amount for loans of over $1200 { c, a | l ( c, l borrower b( l, b, a loan b = “Perryridge”))} or { c, a | l ( c, l borrower l, “Perryridge”, a loan)} Find the names of all customers who have a loan from the Perryridge branch and the loan amount: { c | l, b, a ( c, l borrower l, b, a loan a > 1200)} Find the names of all customers who have a loan of over $1200 { l, b, a | l, b, a loan a > 1200}
63.
©Silberschatz, Korth and
Sudarshan3.63Database System Concepts Example Queries Find the names of all customers having a loan, an account, or both at the Perryridge branch: { c | s, n ( c, s, n customer) x,y,z( x, y, z branch y = “Brooklyn”) a,b( x, y, z account c,a depositor)} Find the names of all customers who have an account at all branches located in Brooklyn: { c | l ({ c, l borrower b,a( l, b, a loan b = “Perryridge”)) a( c, a depositor b,n( a, b, n account b = “Perryridge”))}
64.
©Silberschatz, Korth and
Sudarshan3.64Database System Concepts Safety of Expressions { x1, x2, …, xn | P(x1, x2, …, xn)} is safe if all of the following hold: 1.All values that appear in tuples of the expression are values from dom(P) (that is, the values appear either in P or in a tuple of a relation mentioned in P). 2.For every “there exists” subformula of the form x (P1(x)), the subformula is true if and only if there is a value of x in dom(P1) such that P1(x) is true. 3. For every “for all” subformula of the form x (P1 (x)), the subformula is true if and only if P1(x) is true for all values x from dom (P1).
65.
End of Chapter
3
66.
©Silberschatz, Korth and
Sudarshan3.66Database System Concepts Result of s branch-name = “Perryridge” (loan)
67.
©Silberschatz, Korth and
Sudarshan3.67Database System Concepts Loan Number and the Amount of the Loan
68.
©Silberschatz, Korth and
Sudarshan3.68Database System Concepts Names of All Customers Who Have Either a Loan or an Account
69.
©Silberschatz, Korth and
Sudarshan3.69Database System Concepts Customers With An Account But No Loan
70.
©Silberschatz, Korth and
Sudarshan3.70Database System Concepts Result of borrower loan
71.
©Silberschatz, Korth and
Sudarshan3.71Database System Concepts Result of s branch-name = “Perryridge” (borrower loan)
72.
©Silberschatz, Korth and
Sudarshan3.72Database System Concepts Result of Pcustomer-name
73.
©Silberschatz, Korth and
Sudarshan3.73Database System Concepts Result of the Subexpression
74.
©Silberschatz, Korth and
Sudarshan3.74Database System Concepts Largest Account Balance in the Bank
75.
©Silberschatz, Korth and
Sudarshan3.75Database System Concepts Customers Who Live on the Same Street and In the Same City as Smith
76.
©Silberschatz, Korth and
Sudarshan3.76Database System Concepts Customers With Both an Account and a Loan at the Bank
77.
©Silberschatz, Korth and
Sudarshan3.77Database System Concepts Result of Pcustomer-name, loan-number, amount (borrower loan)
78.
©Silberschatz, Korth and
Sudarshan3.78Database System Concepts Result of Pbranch-name(scustomer-city = “Harrison”(customer account depositor))
79.
©Silberschatz, Korth and
Sudarshan3.79Database System Concepts Result of Pbranch-name(sbranch-city = “Brooklyn”(branch))
80.
©Silberschatz, Korth and
Sudarshan3.80Database System Concepts Result of Pcustomer-name, branch-name(depositor account)
81.
©Silberschatz, Korth and
Sudarshan3.81Database System Concepts The credit-info Relation
82.
©Silberschatz, Korth and
Sudarshan3.82Database System Concepts Result of Pcustomer-name, (limit – credit-balance) as credit-available(credit-info).
83.
©Silberschatz, Korth and
Sudarshan3.83Database System Concepts The pt-works Relation
84.
©Silberschatz, Korth and
Sudarshan3.84Database System Concepts The pt-works Relation After Grouping
85.
©Silberschatz, Korth and
Sudarshan3.85Database System Concepts Result of branch-name sum(salary) (pt-works)
86.
©Silberschatz, Korth and
Sudarshan3.86Database System Concepts Result of branch-name sum salary, max(salary) as max-salary (pt-works)
87.
©Silberschatz, Korth and
Sudarshan3.87Database System Concepts The employee and ft-works Relations
88.
©Silberschatz, Korth and
Sudarshan3.88Database System Concepts The Result of employee ft-works
89.
©Silberschatz, Korth and
Sudarshan3.89Database System Concepts The Result of employee ft-works
90.
©Silberschatz, Korth and
Sudarshan3.90Database System Concepts Result of employee ft-works
91.
©Silberschatz, Korth and
Sudarshan3.91Database System Concepts Result of employee ft-works
92.
©Silberschatz, Korth and
Sudarshan3.92Database System Concepts Tuples Inserted Into loan and borrower
93.
©Silberschatz, Korth and
Sudarshan3.93Database System Concepts Names of All Customers Who Have a Loan at the Perryridge Branch
94.
©Silberschatz, Korth and
Sudarshan3.94Database System Concepts E-R Diagram
95.
©Silberschatz, Korth and
Sudarshan3.95Database System Concepts The branch Relation
96.
©Silberschatz, Korth and
Sudarshan3.96Database System Concepts The loan Relation
97.
©Silberschatz, Korth and
Sudarshan3.97Database System Concepts The borrower Relation
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