The Relational Model Relations A relation is a

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The Relational Model

The Relational Model

Relations A relation is a more concrete construction, of something we have seen before,

Relations A relation is a more concrete construction, of something we have seen before, the ER diagram. A relation is (just!) a table! We will use table and relation interchangeably, except where there is a possibility of confusion. S. S. N name street city students name S. S. N street city Lisa 1272 Blaine Riverside Bart 5592 Apple Irvine Lisa 7552 11 th Riverside Sue 5555 Main Oceanside The students relation

A relation consists of a relational schema and a relational instance. A relation schema

A relation consists of a relational schema and a relational instance. A relation schema is essentially a list of column names with their data types. In this case… students(name : string, S. S. N : string, street : string, city : string) • An relation instance is made up of zero of more tuples (rows, records) name S. S. N street city Lisa 1272 Blaine Riverside Bart 5592 Apple Irvine Lisa 7552 11 th Riverside Sue 5555 Main Oceanside

A schema specifies a relation’s name. students(name : string, S. S. N : string,

A schema specifies a relation’s name. students(name : string, S. S. N : string, street : string, city : string) A schema also specifies the name of each field, and its domain. Fields are often referred to as columns, attributes, dimensions

A minor, but important point about relations, they are unordered. name S. S. N

A minor, but important point about relations, they are unordered. name S. S. N street city name S. S. N city street Lisa 1272 Blaine Riverside Lisa 1272 Riverside Blaine Bart 5592 Apple Irvine Bart 5592 Irvine Apple Lisa 7552 11 th Riverside Lisa 7552 Riverside 11 th Sue 5555 Main Oceanside Sue 5555 Oceanside Main This is not a problem, since we refer to fields by name. However sometimes, we refer to the fields by their column number, in which case the ordering becomes important. I will point this out when we get there. Also, the tuples are unordered too!

Note that every tuple in our instance is unique. This is not a coincidence.

Note that every tuple in our instance is unique. This is not a coincidence. The definition of relation demands it. Later we will see how we can represent weak entities in relations. name S. S. N street city Lisa 1272 Blaine Riverside Bart 5592 Apple Irvine Lisa 7552 11 th Riverside Sue 5592 Main Oceanside

The number of fields is called the degree (or arity, or dimensionality of the

The number of fields is called the degree (or arity, or dimensionality of the relation). Below we have a table of degree 4. The number of tuples cardinality of the relation Of course, we don’t count the row that has the labels! To the right we have a table of cardinality 3. name S. S. N street city Lisa 1272 Blaine Riverside Bart 5592 Apple Irvine Lisa 7552 11 th Riverside

students(name : string, S. S. N : string, street : string, city : string)

students(name : string, S. S. N : string, street : string, city : string) Note that relations have primary keys, just like ER diagrams. Remember that the primary key might not be one field, it may be a combination of two or more fields. name S. S. N street city Lisa 1272 Blaine Riverside Bart 5592 Apple Irvine Lisa 7552 11 th Riverside Sue 5555 Main Oceanside

Translating ER diagrams into Relations We need to figure out how to translate ER

Translating ER diagrams into Relations We need to figure out how to translate ER diagrams into relations. There are only three cases to worry about. • Strong entity sets • Weak entity sets • Relationship sets Name Number Course days Teaches Name PID Professor

 • Strong entity sets days Name Number Course Name PID Teaches Professor professor(PID

• Strong entity sets days Name Number Course Name PID Teaches Professor professor(PID : string, name : string) This is trivial, the primary key of the ER diagram becomes the primary key of the relation. All other fields are copied in (in any order) PID name 1234 3421 2342 4531 Keogh Lee Smyth Lee

 • Weak entity sets days Name Number Teaches Course Name PID Professor course(PID

• Weak entity sets days Name Number Teaches Course Name PID Professor course(PID : string, number : string, name : string) PID number name 1234 3421 2342 4531 CS 12 CS 15 C++ Java C++ LISP 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. The “imported” key from the strong entity set is called the foreign key. All other fields are copied in (in any order)

 • Relationship entity sets Name Number days Teaches Course Name PID Professor teaches(PID

• Relationship entity sets Name Number days Teaches Course Name PID Professor teaches(PID : string, days : string ) 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 • For the other cases, the relation’s primary key is taken from the strong entity set. PID days 1234 3421 2342 4531 mwf wed tue sat

So, this ER Model… Name Number Course days Name PID Teaches Professor … maps

So, this ER Model… Name Number Course days Name PID Teaches Professor … maps to this database schema professor(PID : string, name : string) course(PID : string, number : string, name : string) teaches(PID : string, days : string)

We have seen how to create a database schema, how do we create an

We have seen how to create a database schema, how do we create an actual database on our computers? professor(PID : string, name : string) course(PID : string, number : string, name : string) teaches(PID : string, days : string)

…how do we create an actual database on our computers? We use SQL, a

…how do we create an actual database on our computers? We use SQL, a language that allows us to build, modify and query databases. professor(PID : string, name : string)

SQL (Structured Query Language) • SQL is a language that allows us to build,

SQL (Structured Query Language) • SQL is a language that allows us to build, modify and query databases. • SQL is an ANSI standard language. American National Standards Institute • SQL is the “engine” behind Oracle, Sybase, Microsoft SQL Server, Informix, Access, Ingres, etc. • Most of these systems have build GUIs on top of the command line interface, so you don’t normally write statements directly in SQL (although you can).

Important Note • In our textbook, the authors introduce SQL at the same time

Important Note • In our textbook, the authors introduce SQL at the same time as they introduce the relational model (Chapter 3). • My plan is a little different. I plan to discuss operations on databases (using relational algebra) in a more abstract way, and revisit SQL later in the course. • I encourage you to glance at the SQL material as you read about the relational model in Chapter 3, but don’t worry about the details of SQL just yet.

Relational Algebra • Procedural language • Five basic operators • selection • projection •

Relational Algebra • Procedural language • Five basic operators • selection • projection • union • set difference • Cross product SQL is closely based on relational algebra. select project (why no intersection? ) difference Cartesian product • The are some other operators which are composed of the above operators. These show up so often that we give them special names. • The operators take one or two relations as inputs and give a new relation as a result.

Select Operation – Example • Relation r • A=B ^ D > 5 (r)

Select Operation – Example • Relation r • A=B ^ D > 5 (r) lowercase Greek sigma A B C D 1 7 5 7 12 3 23 10 A B C D 1 7 23 10 Intuition: The select operation allows us to retrieve some rows of a relation (by “some” I mean anywhere from none of them to all of them) Here I have retrieved all the rows of the relation r where either the value in field A equals the value in field B, or the value in field D is greater than 5.

Select Operation • Notation: p(r) lowercase Greek sigma • p is called the selection

Select Operation • Notation: p(r) lowercase Greek sigma • 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: (professor)

Project Operation – Example I • Relation r: • A, C (r) Greek capital

Project Operation – Example I • Relation r: • A, C (r) Greek capital letter pi A B C 10 7 20 1 30 1 40 2 A C 7 1 1 2 Intuition: The project operation allows us to retrieve some columns of a relation (by “some” I mean anywhere from none of them to all of them) Here I have retrieved columns A and C.

Project Operation – Example II • Relation r: • A, C (r) A B

Project Operation – Example II • Relation r: • A, C (r) A B C 10 1 20 1 30 1 40 2 Intuition: The project operation removes duplicate rows, since relations are sets. A C 1 1 1 2 2 = Here there are two rows with A = and C = 1. So one was discarded.

Project Operation • Notation: A 1, A 2, …, Ak (r) Greek capital letter

Project Operation • Notation: A 1, A 2, …, Ak (r) Greek capital letter pi where A 1, A 2 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.

Union Operation – Example Relations r, s: A B 1 2 2 3 1

Union Operation – Example Relations r, s: A B 1 2 2 3 1 s r r s: A B 1 2 1 3 Intuition: The union operation concatenates two relations, and removes duplicate rows (since relations are sets). Here there are two rows with A = and B = 2. So one was discarded.

Union Operation • Notation: r s • Defined as: r s = {t |

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. , 2 nd column of r deals with the same type of values as does the 2 nd column of s). Although the field types must be the same, the names can be different. For example I can union professor and lecturer where: professor(PID : string, name : string) lecturer(LID : string, first_name : string)

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

Set Difference Operation – Example Relations r, s: A B 1 2 2 3 1 s r r – s: A B 1 1 Intuition: The set difference operation returns all the rows that are in r but not in s.

Set Difference Operation • Notation r – s • Defined as: r – s

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 • Note that in general r – s s – r

Cross-Product Operation -Example Relations r, s: A B 1 2 r r x s:

Cross-Product Operation -Example Relations r, s: A B 1 2 r r x s: C D E 10 10 20 10 a a b b s A B C D E 1 1 2 2 10 19 20 10 10 10 20 10 a a b b Intuition: The cross product operation returns all possible combinations of rows in r with rows in s. In other words the result is every possible pairing of the rows of r and s.

Cross-Product Operation-Example Relations r, s: A B 1 2 r r x s: C

Cross-Product Operation-Example Relations r, s: A B 1 2 r r x s: C D E 10 10 20 10 a a b b s A B C D E 1 1 2 2 10 19 20 10 10 10 20 10 a a b b Intuition: The cross product operation returns all possible combinations of rows in r with rows in s. In other words the result is every possible pairing of the rows of r and s.

Cross-Product Operation • Notation r x s • Defined as: r x s =

Cross-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 names of r(R) and s(S) are not disjoint, then renaming must be used.

Composition of Operations • We can build expressions using multiple operations • Example: A=

Composition of Operations • We can build expressions using multiple operations • Example: A= C(r x s) A B 1 2 r “take the cross product of r and s, then return only the rows where A equals B” r x s: C D E 10 10 20 10 a a b b s A=C(r x s) A B C D E 1 1 2 2 10 10 20 10 a a b b A B C D E 1 2 2 10 20 a a b

Rename Operation • Allows us to name, and therefore to refer to, the results

Rename Operation • Allows us to name, and therefore to refer to, the results of relational -algebra expressions. Example: my. Relation (r – s) Take the set difference of r and s, and call the result my. Relation Renaming in relational algebra is essentiality the same as assignment in a programming language A B 1 2 1 A B 2 3 s r A B 1 1 my. Relation

Rename Operation If a relational-algebra expression E has arity n, then A B 1

Rename Operation If a relational-algebra expression E has arity n, then A B 1 2 1 x (A 1, A 2, …, An) (E) returns the result of expression E under the name X, and with the attributes renamed to A 1, A 2, …. , An. Example my. Relation(E, K) (r – s) Take the set difference of r and s, and call the result my. Relation, while renaming the first field E and the second field K. A B 2 3 s r E K 1 1 my. Relation

Banking Examples branch (branch-name, branch-city, assets) customer (customer-name, customer-street, customer-only) account (account-number, branch-name, balance)

Banking Examples 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) Note that I have not indicated primary keys here for simplicity.

Quick note on notation good_customers bad_customers customer-name loan-number Patty 1234 Seymour 3432 Apu 3421

Quick note on notation good_customers bad_customers customer-name loan-number Patty 1234 Seymour 3432 Apu 3421 Marge 3467 Selma 2342 Selma 7625 Ned 4531 Abraham 3597 If we have two or more relations which feature the same attribute names, we could confuse them. To prevent this we can use dot notation. For example good_customers. loan-number

Example Queries • Find all loans of over $1200 amount > 1200 (loan) loan

Example Queries • Find all loans of over $1200 amount > 1200 (loan) loan amount > 1200 (loan) “select from the relation loan, only the rows which have a amount greater than 1200” loan-number 1234 3421 2342 4531 branch-name amount Riverside 1, 923. 03 Irvine 123. 00 Dublin 56. 25 Prague 120. 03 1234 Riverside 1, 923. 03

Example Queries • Find the loan number for each loan of an amount greater

Example Queries • Find the loan number for each loan of an amount greater than $1200 loan-number ( amount > 1200 (loan)) “select from the relation loan, only the rows which have a amount greater than 1200, then project out just the loan_number” loan-number branch-name loan amount > 1200 (loan) loan-number ( amount > 1200 (loan)) amount 1234 Riverside 3421 Irvine 123. 00 2342 Dublin 56. 25 4531 Prague 120. 03 1234 Riverside 1234 1, 923. 03

Example Queries • Find all loans greater than $1200 or less than $75 amount

Example Queries • Find all loans greater than $1200 or less than $75 amount > 1000 or amount < 75(loan) “select from the relation loan, only the rows which have a amount greater than 1000 or an amount less than 75 loan amount > 1000 or amount < 75(loan) loan-number branch-name amount 1234 Riverside 1, 923. 03 3421 Irvine 123. 00 2342 Dublin 56. 25 4531 Prague 120. 03 1234 Riverside 2342 Dublin 1, 923. 03 56. 25

Example Queries • Find the names of all customers who have a loan, an

Example Queries • Find the names of all customers who have a loan, an account, or both, from the bank customer-name (borrower) customer-name (depositor) depositor borrower customer-name loan-number customer-name account-number Patty 1234 Moe 3467 Apu 3421 Apu 2312 Selma 2342 Patty 9999 Ned 4531 Krusty 3423 customer-name (borrower) Moe Apu customer-name (depositor) Patty Moe Apu Krusty Apu Selma Patty Ned Krusty

Example Queries Note this example is split over two slides! Find the names of

Example Queries Note this example is split over two slides! Find the names of all customers who have a loan at the Riverside branch. customer-name ( branch-name=“Riverside ” ( borrower. loan-number = loan-number(borrower x loan))) borrower We retrieve borrower and loan… …we calculate their cross product… loan customer-name loan-number branch-name amount Patty 1234 Riverside Apu 3421 Irvine customer-name borrower. loan-number branch-name Patty 1234 Riverside Patty 1234 3421 Irvine Apu 3421 1234 Riverside Apu 3421 Irvine 1, 923. 03 123. 00 amount 1, 923. 03 123. 00

 customer-name ( branch-name=“Riverside ” ( borrower. loan-number = loan-number(borrower x loan))) …we calculate

customer-name ( branch-name=“Riverside ” ( borrower. loan-number = loan-number(borrower x loan))) …we calculate their cross product… …we select the rows where borrower. loannumber is equal to loan-number… …we select the rows where branch-name is equal to customer-name borrower. loan-number branch-name Patty 1234 Riverside Patty 1234 3421 Irvine Apu 3421 1234 Riverside Apu 3421 Irvine customer-name borrower. loan-number branch-name Patty 1234 Riverside “Riverside” …we project out the customer-name. Patty 1234 amount 1, 923. 03 123. 00 amount 1, 923. 03

Example Queries Note this example is split over three slides! Find the largest account

Example Queries Note this example is split over three slides! Find the largest account balance. . . we will need to rename account relation as d. . . balance(account) - account. balance( account. balance < d. balance (account x d (account))) d account We do a rename to get a “copy” of account which we call d… … next we will do a cross product… account- balance number Apu 100. 30 Patty 12. 34 Lenny 45. 34

 balance(account) - account. balance( account. balance < d. balance (account x d (account)))

balance(account) - account. balance( account. balance < d. balance (account x d (account))) accountnumber … do a cross product… …select out all rows where account. balance is less than d. balance… . . next we project… account. balance d. accountnumber d. balance Apu 100. 30 Patty 12. 34 Apu 100. 30 Lenny 45. 34 Patty 12. 34 Apu 100. 30 Patty 12. 34 Lenny 45. 34 Apu 100. 30 Lenny 45. 34 Patty 12. 34 Lenny 45. 34 accountnumber account. balance d. accountnumber Patty 12. 34 Apu Patty 12. 34 Lenny 45. 34 Apu d. balance 100. 30 45. 34 100. 30

 balance(account) - account. balance( account. balance < d. balance (account x d (account)))

balance(account) - account. balance( account. balance < d. balance (account x d (account))) accountnumber . . next we project out account. balance… …then we do a set difference between it an the original account. balance from the account relation… … the set difference leaves us with one number, the largest value! account. balance d. accountnumber Patty 12. 34 Apu Patty 12. 34 Lenny 45. 34 Apu account. balance 12. 34 account- balance number Apu 100. 30 Patty 12. 34 Lenny 45. 34 100. 30 12. 34 45. 34 d. balance 100. 30 45. 34 100. 30

Formal Definition • A basic expression in the relational algebra consists of either one

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 E 1 and E 2 be relational-algebra expressions; the following are all relational-algebra expressions: – – – E 1 E 2 E 1 - E 2 E 1 x E 2 p (E 1), P is a predicate on attributes in E 1 s(E 1), S is a list consisting of some of the attributes in E 1 – x (E 1), x is the new name for the result of E 1

Additional Operations We define additional operations that do not add any power to the

Additional Operations We define additional operations that do not add any power to the relational algebra, but that simplify common queries. – Natural join – Conditional Join – Equi join – Division – Set intersection All joins are really special cases of conditional join

Natural-Join Operation: Motivation Very often we have a query and the answer is not

Natural-Join Operation: Motivation Very often we have a query and the answer is not contained in a single relation. For example, I might wish to know where Apu banks. The classic relational algebra way to do such queries is a cross product, followed by a selection which tests for equality on some pair of fields. borrower cust-name l-number branch Patty 1234 Dublin Apu 3421 Irvine borrower. l-number = loan. l-number(borrower x loan))) While this works… • it is unintuitive • it requires a lot of memory • the notation is cumbersome loan cust-name borrower. l-number loan. l-number branch Patty 1234 Dublin Patty 1234 3421 Irvine Apu 3421 1234 Dublin Apu 3421 Irvine cust-name borrower. l-number loan. l-number branch Patty 1234 Dublin Apu 3421 Irvine Note that is this example the two relations are the same size (2 by 2), this does not have to be the case. So we have a more intuitive way of achieving the same effect, the natural join, denoted by the symbol

Natural-Join Operation: Intuition Natural join combines a cross product and a selection into one

Natural-Join Operation: Intuition Natural join combines a cross product and a selection into one operation. It performs a selection forcing equality on those attributes that appear in both relation schemes. Duplicates are removed as in all relation operations. So if the relations have one attribute in common, as in the last slide (“l-number”), for example, we have… borrower loan = borrower. l-number = loan. l-number(borrower x loan))) There are two special cases: • If the two relations have no attributes in common, then their natural join is simply their cross product. • If the two relations have more than one attribute in common, then the natural join selects only the rows where all pairs of matching attributes match. (lets see an example on the next slide).

A l-name f-name age Bouvier Selma 40 Bouvier Patty 40 Smith Maggie 2 Both

A l-name f-name age Bouvier Selma 40 Bouvier Patty 40 Smith Maggie 2 Both the l-name and the f-name match, so select. Only the f-names match, so don’t select. Only the l-names match, so don’t select. We remove duplicate attributes… The natural join of A and B Note that this is just a way to visualize the natural join, we don’t really have to do the cross product as in this example B l-name f-name ID Bouvier Selma 1232 Smith Selma 4423 l-name f-name age l-name f-name ID Bouvier Selma 40 Bouvier Selma 1232 Bouvier Patty 40 Smith Selma 4423 Smith Maggie 2 Bouvier Selma 1232 Bouvier Selma 40 Smith Selma 4423 Bouvier Patty 40 Bouvier Selma 1232 Smith Maggie 2 Smith Selma 4423 l-name f-name age l-name f-name ID Bouvier Selma 40 Bouvier Selma 1232 A B= l-name f-name age ID Bouvier Selma 40 1232

Natural-Join Operation • Notation: r s • Let r and s be relations on

Natural-Join Operation • Notation: r s • Let r and s be relations on schemas R and S respectively. The result is a relation on schema R S which is obtained by considering 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, a tuple t is added 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 • Relations r, s: A B C D B

Natural Join Operation – Example • Relations r, s: A B C D B D E 1 2 4 1 2 a a b 1 3 1 2 3 a a a b b r r s s A B C D E 1 1 2 a a b How did we get here? Lets do a trace over the next few slides… Warning! Example spread over many slides, you may wish to edit before printing.

A B C D B D E 1 2 4 1 2 a a

A B C D B D E 1 2 4 1 2 a a b 1 3 1 2 3 a a a b b r s First we note which attributes the two relations have in common…

A B C D B D E 1 2 4 1 2 a a

A B C D B D E 1 2 4 1 2 a a b 1 3 1 2 3 a a a b b r s A B C D E 1 1 a a There are two rows in s that match our first row in r, (in the relevant attributes) so both are joined to our first row…

A B C D B D E 1 2 4 1 2 a a

A B C D B D E 1 2 4 1 2 a a b 1 3 1 2 3 a a a b b r s A B C D E 1 1 a a …there are no rows in s that match our second row in r, so do nothing…

A B C D B D E 1 2 4 1 2 a a

A B C D B D E 1 2 4 1 2 a a b 1 3 1 2 3 a a a b b r s A B C D E 1 1 a a …there are no rows in s that match our third row in r, so do nothing…

A B C D B D E 1 2 4 1 2 a a

A B C D B D E 1 2 4 1 2 a a b 1 3 1 2 3 a a a b b r s A B C D E 1 1 a a There are two rows in s that match our fourth row in r, so both are joined to our fourth row…

A B C D B D E 1 2 4 1 2 a a

A B C D B D E 1 2 4 1 2 a a b 1 3 1 2 3 a a a b b r s A B C D E 1 1 2 a a b There is one row that matches our fifth row in r, . . so it is joined to our fifth row and we are done!

Conditional-Join Operation: The conditional join is actually the most general type of join. I

Conditional-Join Operation: The conditional join is actually the most general type of join. I introduced the natural join first only because it is more intuitive and. . natural! Just like natural join, conditional join combines a cross product and a selection into one operation. However instead of only selecting rows that have equality on those attributes that appear in both relation schemes, we allow selection based on any predicate. r c s = c(r x s) Where c is any predicate the attributes of r and/or s Duplicate rows are removed as always, but duplicate columns are not removed!

Conditional-Join Example: We want to find all women that are older than their husbands…

Conditional-Join Example: We want to find all women that are older than their husbands… r l-name f-name marr-Lic age l-name Simpson Marge 777 35 Simpson Homer Lovejoy Helen 234 38 Flanders Maude 555 24 Krabappel Edna r 978 s Lovejoy f-name marr-Lic age 777 36 Timothy 234 36 Simpson Bart null 9 40 r. age > s. age AND r. Marr-Lic = r. Marr-Lic s r. l-name r. f-name r. Marr-Lic r. age s. l-name s. f-name s. marr-Lic s. age Lovejoy Helen 234 38 Timothy 234 36 Lovejoy Note we have removed ambiguity of attribute names by using “dot” notation Also note the redundant information in the marr-lic attributes

Set-Intersection Operation - Example Relation r, s: A B 1 2 1 A 2

Set-Intersection Operation - Example Relation r, s: A B 1 2 1 A 2 3 s r r s B A B 2 Intuition: The intersection operation returns all the rows that are in both r and s.

Set-Intersection Operation • • Notation: r s Defined as: r s ={ t |

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)

r /s Division Operation • Suited to queries that include the phrase “for all”.

r /s Division Operation • Suited to queries that include the phrase “for all”. • Let r and s be relations on schemas R and S respectively where – R = (A 1, …, Am, B 1, …, Bn) – S = (B 1, …, Bn) The result of r / s is a relation on schema R – S = (A 1, …, Am) r / s = { t | t R-S(r) u s ( tu r ) }

Division Operation – Example Relations r, s: A B B 1 2 3 1

Division Operation – Example Relations r, s: A B B 1 2 3 1 1 1 3 4 6 1 2 s r occurs in the presence of both 1 and 2, so it is returned. g does not occur in the presence of both 1 and 2, so is ignored. . A r / s:

Another Division Example Relations r, s: A B C D E a a a

Another Division Example Relations r, s: A B C D E a a a a a a b a b b 1 1 3 1 1 1 a b 1 1 r s r /s: A B C a a < , a , > occurs in the presence of both <a, 1> and <b, 1>, so it is returned. < , a , > does not occur in the presence of both <a, 1> and <b, 1>, so it is ignored.

Assignment Operation • The assignment operation ( ) provides a convenient way to express

Assignment Operation • The assignment operation ( ) provides a convenient way to express complex queries, write query as a sequential program consisting of a series of assignments followed by an expression whose value is displayed as a result of the query. • Assignment must always be made to a temporary relation variable. • Example: Write r s as temp 1 R-S (r) temp 2 R-S ((temp 1 x s) – R-S, S (r)) result = temp 1 – temp 2 – The result to the right of the is assigned to the relation variable on the left of the . – May use variable in subsequent expressions.

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

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

Generalized Projection • Extends the projection operation by allowing arithmetic functions to be used

Generalized Projection • Extends the projection operation by allowing arithmetic functions to be used in the projection list. F 1, F 2, …, Fn(E) • E is any relational-algebra expression • Each of F 1, F 2, …, Fn are arithmetic expressions involving constants and attributes in the schema of E. • Given relation credit-info(customer-name, limit, creditbalance), find how much more each person can spend: customer-name, limit – credit-balance (credit-info)

Generalized Projection Given relation credit-info(customer-name, limit, creditbalance), find how much more each person can

Generalized Projection Given relation credit-info(customer-name, limit, creditbalance), find how much more each person can spend: customer-name, limit – credit-balance (credit-info) customer-name credit-info limit credit-balance Simpson, Marge 500 400 Lovejoy, Helen 2000 1500 Flanders, Maude 0 0 Krabappel, Edna 50 11 100 500 0 39

Aggregate Functions and Operations • Aggregation function takes a collection of values and returns

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 G 1, G 2, …, Gn – – g F 1( A 1), F 2( A 2), …, Fn( An) (E) E is any relational-algebra expression G 1, G 2 …, Gn is a list of attributes on which to group (can be empty) Each Fi is an aggregate function (i. e avg, min, max etc) Each Ai is an attribute name

Aggregate Operation – Example • Relation r: g sum(c) (r) A B C 7

Aggregate Operation – Example • Relation r: g sum(c) (r) A B C 7 7 3 10 sum-C 27 i. e we want to find the sum of all the numbers in attribute C

Aggregate Operation – Example Relation account grouped by last-name: account i. e calculate the

Aggregate Operation – Example Relation account grouped by last-name: account i. e calculate the total balances, grouped by lastname. last-name account-number balance Simpson Flanders Nahasapeemapetilon A-102 A-201 A-217 A-215 A-222 400 900 750 11700 g sum(balance) (account) last-name balance Simpson Flanders Nahasapeemapetilon 1300 1500 11700 Yes yes, I make good money, but I was shot 14 times last year

Outer Join • An extension of the join operation that avoids loss of information.

Outer Join • An extension of the join operation that avoids loss of information. • Computes the join and then adds tuples from 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 • Relation loan n Relation borrower loan-number branch-name L-170 L-230

Outer Join – Example • Relation loan n Relation borrower loan-number branch-name L-170 L-230 L-260 Springfield Shelbyville Dublin amount 3000 4000 1700 customer-name loan-number Simpson Wiggum Flanders L-170 L-230 L-155

Outer Join – Example • Inner Join loan-number branch-name L-170 L-230 Springfield Shelbyville amount

Outer Join – Example • Inner Join loan-number branch-name L-170 L-230 Springfield Shelbyville amount 3000 4000 customer-name Simpson Wiggum Borrower • Left Outer Join loan borrower loan-number L-170 L-230 L-260 branch-name Springfield Shelbyville Dublin amount 3000 4000 1700 customer-name Simpson Wiggum null

Outer Join – Example loan-number branch-name L-170 L-230 L-155 Springfield Shelbyville null loan-number branch-name

Outer Join – Example loan-number branch-name L-170 L-230 L-155 Springfield Shelbyville null loan-number branch-name L-170 L-230 L-260 L-155 Springfield Shelbyville Dublin null Right Outer Join loan 3000 4000 null customer-name Simpson Wiggum Flanders borrower Full Outer Join loan amount borrower amount 3000 4000 1700 null customer-name Simpson Wiggum null Flanders

Null Values • It is possible for tuples to have a null value, denoted

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 • Comparisons with null values return the special truth value unknown –

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