CA II,U3 A,RELATIONAL ALGEBRA

Slide 1: 

UNIT III A RELATIONAL ALGEBRA COMPUTER APPLICATION II M.S.TUFAIL DEPARTMENT OF MECHANICAL ENGG. Y.C.C.E, NAGPUR

UNIT 3A: Relational Model : 

UNIT 3A: Relational Model Structure of Relational Databases Relational Algebra Tuple Relational Calculus Domain Relational Calculus Extended Relational-Algebra-Operations Modification of the Database Views

Example of a Relation : 

Example of a Relation

About Relational Model : 

About Relational Model Order of tuples not important Order of attributes not important (in theory) Collection of relation schemas (intension) Relational database schema Corresponding relation instances (extension) Relational database intension vs. extension schema vs. data metadata includes schema

Why Relations? : 

Why Relations? Very simple model. Often a good match for the way we think about our data. Abstract model that underlies SQL, the most important language in DBMS’s today. But SQL uses “bags” while the abstract relational model is set-oriented.

Relational Design : 

Relational Design Simplest approach (not always best): convert each E.S. to a relation and each relationship to a relation. Entity Set  Relation E.S. attributes become relational attributes. Becomes: Beers(name, manf) Beers name manf

Relational Model : 

Relational Model Table = relation. Column headers = attributes. Row = tuple Beers Relation schema = name(attributes) + other structure info.,e.g., keys, other constraints. Example: Beers(name, manf) Order of attributes is arbitrary, but in practice we need to assume the order given in the relation schema. Relation instance is current set of rows for a relation schema. Database schema = collection of relation schemas. name manf WinterBrew Pete’s BudLite A.B. … …

Basic Structure : 

Basic Structure Formally, given sets D1, D2, …. Dn a relation r is a subset of D1 x D2 x … x DnThus a relation is a set of n-tuples (a1, a2, …, an) where each ai  Di Example: if customer-name = {Jones, Smith, Curry, Lindsay} customer-street = {Main, North, Park} customer-city = {Harrison, Rye, Pittsfield}Then r = { (Jones, Main, Harrison), (Smith, North, Rye), (Curry, North, Rye), (Lindsay, Park, Pittsfield)} is a relation over customer-name x customer-street x customer-city

Relational Data Model : 

A1 A2 A3 ... An a1 a2 a3 an b1 b2 a3 cn a1 c3 b3 bn . . . x1 v2 d3 wn Relational Data Model Set theoretic Domain — set of values like a data type Cartesian product (or product) D1 D2 ...  Dn n-tuples (V1,V2,...,Vn) s.t., V1 D1, V2 D2,...,Vn Dn Relation-subset of cartesian product of one or more domains FINITE only; empty set allowed Tuples = members of a relation inst. Arity = number of domains Components = values in a tuple Domains — corresp. with attributes Cardinality = number of tuples Relation as table Rows = tuples Columns = components Names of columns = attributes Set of attribute names = schema REL (A1,A2,...,An) Arity C a r d i n a l i t y Attributes Component Tuple

Relation: Example : 

Name address tel # 5 3 7 Cardinality of domain Domains N A T N1 A1 T1 N2 A2 T2 N3 A3 T3 N4 T4 N5 T5 T6 T7 Relation: Example Domain of Relation N A T N1 A1 T1 N1 A1 T2 N1 A1 T3 . . . N1 A1 T7 N1 A2 T1 N1 A3 T1 N2 A1 T1 Arity 3 Cardinality <=5x3x7 of relation Tuple µ Domain Component Attribute

Attribute Types : 

Attribute Types Each attribute of a relation has a name The set of allowed values for each attribute is called the domain of the attribute Attribute values are (normally) required to be atomic, that is, indivisible E.g. multivalued attribute values are not atomic E.g. composite attribute values are not atomic The special value null is a member of every domain The null value causes complications in the definition of many operations we shall ignore the effect of null values in our main presentation and consider their effect later

Relation Schema : 

Relation Schema A1, A2, …, An are attributes R = (A1, A2, …, An ) is a relation schema E.g. Customer-schema = (customer-name, customer-street, customer-city) r(R) is a relation on the relation schema R E.g. customer (Customer-schema)

Relation Instance : 

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)

Relation Instance : 

Relation Instance Name Address Telephone Bob 123 Main St 555-1234 Bob 128 Main St 555-1235 Pat 123 Main St 555-1235 Harry 456 Main St 555-2221 Sally 456 Main St 555-2221 Sally 456 Main St 555-2223 Pat 12 State St 555-1235

Relations are Unordered : 

Relations are Unordered Order of tuples is irrelevant (tuples may be stored in an arbitrary order) E.g. account relation with unordered tuples

Database : 

Database A database consists of multiple relations Information about an enterprise is broken up into parts, with each relation storing one part of the information E.g.: account : stores information about accounts depositor : stores information about which customer owns which account customer : stores information about customers Storing all information as a single relation such as bank(account-number, balance, customer-name, ..)results in repetition of information (e.g. two customers own an account) the need for null values (e.g. represent a customer without an account) Normalization theory (Chapter 7) deals with how to design relational schemas

The customer Relation : 

The customer Relation

The depositor Relation : 

The depositor Relation

E-R Diagram for the Banking Enterprise : 

E-R Diagram for the Banking Enterprise

Keys : 

Keys Let K  R K is a superkey of R if values for K are sufficient to identify a unique tuple of each possible relation r(R) by “possible r” we mean a relation r that could exist in the enterprise we are modeling. Example: {customer-name, customer-street} and {customer-name} are both superkeys of Customer, if no two customers can possibly have the same name. K is a candidate key if K is minimalExample: {customer-name} is a candidate key for Customer, since it is a superkey (assuming no two customers can possibly have the same name), and no subset of it is a superkey.

Determining Keys from E-R Sets : 

Determining Keys from E-R Sets Strong entity set. The primary key of the entity set becomes the primary key of the relation. Weak entity set. The primary key of the relation consists of the union of the primary key of the strong entity set and the discriminator of the weak entity set. Relationship set. The union of the primary keys of the related entity sets becomes a super key of the relation. For binary many-to-one relationship sets, the primary key of the “many” entity set becomes the relation’s primary key. For one-to-one relationship sets, the relation’s primary key can be that of either entity set. For many-to-many relationship sets, the union of the primary keys becomes the relation’s primary key

Schema Diagram for the Banking Enterprise : 

Schema Diagram for the Banking Enterprise

Query Languages : 

Query Languages Language in which user requests information from the database. Categories of languages procedural non-procedural “Pure” languages: Relational Algebra Tuple Relational Calculus Domain Relational Calculus Pure languages form underlying basis of query languages that people use.

Relational Algebra : 

Relational Algebra Procedural language Six basic operators select project union set difference Cartesian product rename The operators take two or more relations as inputs and give a new relation as a result.

Select Operation – Example : 

Select Operation – Example Relation r A B C D         1 5 12 23 7 7 3 10 A=B ^ D > 5 (r) A B C D     1 23 7 10

Select Operation : 

Select Operation Notation:  p(r) p is called the selection predicate Defined as: p(r) = {t | t  r and p(t)} Where p is a formula in propositional calculus consisting of terms connected by :  (and),  (or),  (not)Each term is one of: <attribute> op <attribute> or <constant> where op is one of: =, , >, . <.  Example of selection:  branch-name=“Perryridge”(account)

Project Operation – Example : 

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)

Project Operation : 

Project Operation Notation: A1, A2, …, Ak (r) where A1, A2 are attribute names and r is a relation name. The result is defined as the relation of k columns obtained by erasing the columns that are not listed Duplicate rows removed from result, since relations are sets E.g. To eliminate the branch-name attribute of account account-number, balance (account)

Union Operation – Example : 

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

Union Operation : 

Union Operation Notation: r  s Defined as: r  s = {t | t  r or t  s} For r  s to be valid. 1. r, s must have the same arity (same number of attributes) 2. The attribute domains must be compatible (e.g., 2nd column of r deals with the same type of values as does the 2nd column of s) E.g. to find all customers with either an account or a loan customer-name (depositor)  customer-name (borrower)

Set Difference Operation – Example : 

Set Difference Operation – Example Relations r, s: r – s: A B    1 2 1 A B   2 3 r s A B   1 1

Set Difference Operation : 

Set Difference Operation Notation r – s Defined as: r – s = {t | t  r and t  s} Set differences must be taken between compatible relations. r and s must have the same arity attribute domains of r and s must be compatible

Cartesian-Product Operation-Example : 

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 b r s

Cartesian-Product Operation : 

Cartesian-Product Operation Notation r x s Defined as: r x s = {t q | t  r and q  s} Assume that attributes of r(R) and s(S) are disjoint. (That is, R  S = ). If attributes of r(R) and s(S) are not disjoint, then renaming must be used.

Composition of Operations : 

Composition of Operations Can build expressions using multiple operations Example: A=C(r x s) r x s A=C(r x s) 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 A B C D E    1 2 2    10 20 20 a a b

Rename Operation : 

Rename Operation Allows us to name, and therefore to refer to, the results of relational-algebra expressions. Allows us to refer to a relation by more than one name. Example:  x (E) returns the expression E under the name X If a relational-algebra expression E has arity n, then x (A1, A2, …, An) (E) returns the result of expression E under the name X, and with the attributes renamed to A1, A2, …., An.

Banking Example : 

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)

Example Queries : 

Example Queries Find all loans of over $1200 Find the loan number for each loan of an amount greater than $1200 amount > 1200 (loan) loan-number (amount > 1200 (loan))

Example Queries : 

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)

Example Queries : 

Example Queries Find the names of all customers who have a loan at the Perryridge branch. Find the names of all customers who have a loan at the Perryridge branch but do not have an account at any branch of the bank. customer-name (branch-name = “Perryridge” (borrower.loan-number = loan.loan-number(borrower x loan))) – customer-name(depositor) customer-name (branch-name=“Perryridge” (borrower.loan-number = loan.loan-number(borrower x loan)))

Example Queries : 

Example Queries Find the names of all customers who have a loan at the Perryridge branch.  Query 2 customer-name(loan.loan-number = borrower.loan-number( (branch-name = “Perryridge”(loan)) x borrower)) Query 1 customer-name(branch-name = “Perryridge” ( borrower.loan-number = loan.loan-number(borrower x loan)))

Example Queries : 

Example Queries Find the largest account balance Rename account relation as d The query is: balance(account) - account.balance (account.balance < d.balance (account x rd (account)))

Formal Definition : 

Formal Definition A basic expression in the relational algebra consists of either one of the following: A relation in the database A constant relation Let E1 and E2 be relational-algebra expressions; the following are all relational-algebra expressions: E1  E2 E1 - E2 E1 x E2 p (E1), P is a predicate on attributes in E1 s(E1), S is a list consisting of some of the attributes in E1  x (E1), x is the new name for the result of E1

Additional Operations : 

Additional Operations We define additional operations that do not add any power to the relational algebra, but that simplify common queries. Set intersection Natural join Division Assignment

Set-Intersection Operation : 

Set-Intersection Operation Notation: r  s Defined as: r  s ={ t | t  r and t  s } Assume: r, s have the same arity attributes of r and s are compatible Note: r  s = r - (r - s)

Set-Intersection Operation - Example : 

Set-Intersection Operation - Example Relation r, s: r  s A B    1 2 1 A B   2 3 r s A B  2

Natural-Join Operation : 

Notation: r s Natural-Join Operation Let r and s be relations on schemas R and S respectively. Then, r s is a relation on schema R  S obtained as follows: Consider each pair of tuples tr from r and ts from s. If tr and ts have the same value on each of the attributes in R  S, add a tuple t to the result, where t has the same value as tr on r t has the same value as ts on s Example: R = (A, B, C, D) S = (E, B, D) Result schema = (A, B, C, D, E) r s is defined as: r.A, r.B, r.C, r.D, s.E (r.B = s.B  r.D = s.D (r x s))

Natural Join Operation – Example : 

Natural Join Operation – Example Relations r, s: A B      1 2 4 1 2 C D      a a b a b B 1 3 1 2 3 D a a a b b E      r A B      1 1 1 1 2 C D      a a a a b E      s

Example Queries : 

Example Queries Find all customers who have an account from at least the “Downtown” and the Uptown” branches.

Example Queries : 

Find all customers who have an account at all branches located in Brooklyn city. Example Queries

Extended Relational-Algebra-Operations : 

Extended Relational-Algebra-Operations Generalized Projection Outer Join Aggregate Functions

Generalized Projection : 

Generalized Projection Extends the projection operation by allowing arithmetic functions to be used in the projection list.  F1, F2, …, Fn(E) E is any relational-algebra expression Each of F1, F2, …, Fn are are arithmetic expressions involving constants and attributes in the schema of E. Given relation credit-info(customer-name, limit, credit-balance), find how much more each person can spend: customer-name, limit – credit-balance (credit-info)

Aggregate Functions and Operations : 

Aggregate Functions and Operations Aggregation function takes a collection of values and returns a single value as a result. avg: average value min: minimum value max: maximum value sum: sum of values count: number of values Aggregate operation in relational algebra G1, G2, …, Gn g F1( A1), F2( A2),…, Fn( An) (E) E is any relational-algebra expression G1, G2 …, Gn is a list of attributes on which to group (can be empty) Each Fi is an aggregate function Each Ai is an attribute name

Aggregate Operation – Example : 

Aggregate Operation – Example Relation r: A B         C 7 7 3 10 g sum(c) (r) sum-C 27

Aggregate Operation – Example : 

Aggregate Operation – Example Relation account grouped by branch-name: branch-name g sum(balance) (account) branch-name account-number balance Perryridge Perryridge Brighton Brighton Redwood A-102 A-201 A-217 A-215 A-222 400 900 750 750 700 branch-name balance Perryridge Brighton Redwood 1300 1500 700

Aggregate Functions (Cont.) : 

Aggregate Functions (Cont.) Result of aggregation does not have a name Can use rename operation to give it a name For convenience, we permit renaming as part of aggregate operation branch-name g sum(balance) as sum-balance (account)

Outer Join : 

Outer Join An extension of the join operation that avoids loss of information. Computes the join and then adds tuples form one relation that does not match tuples in the other relation to the result of the join. Uses null values: null signifies that the value is unknown or does not exist All comparisons involving null are (roughly speaking) false by definition. Will study precise meaning of comparisons with nulls later

Outer Join – Example : 

Outer Join – Example Relation loan Relation borrower

Outer Join – Example : 

Outer Join – Example Inner Joinloan Borrower

Outer Join – Example : 

Outer Join – Example Right Outer Join loan borrower loan borrower Full Outer Join

Null Values : 

Null Values It is possible for tuples to have a null value, denoted by null, for some of their attributes null signifies an unknown value or that a value does not exist. The result of any arithmetic expression involving null is null. Aggregate functions simply ignore null values Is an arbitrary decision. Could have returned null as result instead. We follow the semantics of SQL in its handling of null values For duplicate elimination and grouping, null is treated like any other value, and two nulls are assumed to be the same Alternative: assume each null is different from each other Both are arbitrary decisions, so we simply follow SQL

Null Values : 

Null Values Comparisons with null values return the special truth value unknown If false was used instead of unknown, then not (A < 5) would not be equivalent to A >= 5 Three-valued logic using the truth value unknown: OR: (unknown or true) = true, (unknown or false) = unknown (unknown or unknown) = unknown AND: (true and unknown) = unknown, (false and unknown) = false, (unknown and unknown) = unknown NOT: (not unknown) = unknown In SQL “P is unknown” evaluates to true if predicate P evaluates to unknown Result of select predicate is treated as false if it evaluates to unknown

Modification of the Database : 

Modification of the Database The content of the database may be modified using the following operations: Deletion Insertion Updating All these operations are expressed using the assignment operator.

Deletion : 

Deletion A delete request is expressed similarly to a query, except instead of displaying tuples to the user, the selected tuples are removed from the database. Can delete only whole tuples; cannot delete values on only particular attributes A deletion is expressed in relational algebra by: r  r – E where r is a relation and E is a relational algebra query.

Deletion Examples : 

Deletion Examples Delete all account records in the sadar branch. Delete all accounts at branches located in Needham. Delete all loan records with amount in the range of 0 to 50 loan  loan – amount 0and amount  50 (loan) account  account – branch-name = “sadar” (account)

Insertion : 

Insertion To insert data into a relation, we either: specify a tuple to be inserted write a query whose result is a set of tuples to be inserted in relational algebra, an insertion is expressed by: r  r  E where r is a relation and E is a relational algebra expression. The insertion of a single tuple is expressed by letting E be a constant relation containing one tuple.

Insertion Examples : 

Insertion Examples Insert information in the database specifying that Smith has $1200 in account A-973 at the Perryridge branch. account  account  {(“sadar”, A-973, 1200)} depositor  depositor  {(“sameer”, A-973)}

Updating : 

Updating A mechanism to change a value in a tuple without charging all values in the tuple Use the generalized projection operator to do this task r   F1, F2, …, FI, (r) Each Fi is either the ith attribute of r, if the ith attribute is not updated, or, if the attribute is to be updated Fi is an expression, involving only constants and the attributes of r, which gives the new value for the attribute

Update Examples : 

Update Examples Make interest payments by increasing all balances by 5 percent. Pay all accounts with balances over $10,000 6 percent interest and pay all others 5 percent account   AN, BN, BAL * 1.06 ( BAL  10000 (account))  AN, BN, BAL * 1.05 (BAL  10000 (account))

Views : 

Views In some cases, it is not desirable for all users to see the entire logical model (i.e., all the actual relations stored in the database.) Consider a person who needs to know a customer’s loan number but has no need to see the loan amount. This person should see a relation described, in the relational algebra, by customer-name, loan-number (borrower loan) Any relation that is not of the conceptual model but is made visible to a user as a “virtual relation” is called a view.

View Definition : 

View Definition A view is defined using the create view statement which has the form create view v as <query expression where <query expression> is any legal relational algebra query expression. The view name is represented by v. Once a view is defined, the view name can be used to refer to the virtual relation that the view generates. View definition is not the same as creating a new relation by evaluating the query expression Rather, a view definition causes the saving of an expression; the expression is substituted into queries using the view.

View Examples : 

View Examples Consider the view (named all-customer) consisting of branches and their customers. We can find all customers of the sitabuldi branch by writing: branch-name (branch-name = “sitabuldi” (all-customer))

Tuple Relational Calculus : 

Tuple Relational Calculus A nonprocedural query language, where each query is of the form {t | P (t) } It is the set of all tuples t such that predicate P is true for t t is a tuple variable, t[A] denotes the value of tuple t on attribute A t  r denotes that tuple t is in relation r P is a formula similar to that of the predicate calculus

Predicate Calculus Formula : 

Predicate Calculus Formula 1. Set of attributes and constants 2. Set of comparison operators: (e.g., , , , , , ) 3. Set of connectives: and (), or (v)‚ not () 4. Implication (): x  y, if x if true, then y is true x  y x v y

Domain Relational Calculus : 

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

End of Chapter 3 : 

End of Chapter 3

E-R Diagram : 

E-R Diagram