Relational algebra
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Relational Algebra. Relational Query Languages. Query = “retrieval program” Language examples: Theoretical : Relational Algebra Relational Calculus tuple relational calculus (TRC) domain relational calculus (DRC) Practical SQL (SEQUEL from System R) QUEL (Ingres) Datalog (Prolog-like)

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Relational query languages l.jpg
Relational Query Languages

  • Query = “retrieval program”

  • Language examples:

    • Theoretical:

      • Relational Algebra

      • Relational Calculus

        • tuple relational calculus (TRC)

        • domain relational calculus (DRC)

    • Practical

      • SQL (SEQUEL from System R)

      • QUEL (Ingres)

      • Datalog (Prolog-like)

  • Theoretical QL’s:

    • give semantics to practical QL’s

    • key to understand query optimization in relational DBMSs

Relational algebra3 l.jpg
Relational Algebra

  • Basic operators

    • select ( )

    • project (p )

    • union ( )

    • set difference ( - )

    • cartesian product ( x )

    • rename ( )

  • The operators take one or two relations as inputs and give a new relation as a result.






Example instances l.jpg
Example Instances






Boats(bid, bname, color)

Sailors(sid, sname, rating, age)

Reserves( sid, bid, day)

Projection l.jpg

  • Examples: ;

  • Retains only attributes that are in the “projection list”.

  • Schemaof result:

    • exactly the columns in the projection list, with the same names that they had in the input relation.

  • Projection operator has to eliminate duplicates (How do they arise? Why remove them?)

    • Note: real systems typically don’t do duplicate elimination unless the user explicitly asks for it. (Why not?)

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Selection ()

  • Selects rows that satisfy selection condition.

  • Result is a relation.

    Schemaof result is same as that of the input relation.

  • Do we need to do duplicate elimination?

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  • Notation: p(r)

  • p is called the selection predicate , rcan be the name of a table, or another query

  • Predicate:

    • Simple

      • attr1 = attr2

      • Attr = constant value

        • (also, <, > , etc)

    • Complex

      • predicate1 AND predicate2

      • predicate1 OR predicate2

      • NOT (predicate)

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Union and Set-Difference

  • All of these operations take two input relations, which must beunion-compatible:

    • Same number of columns (attributes).

    • `Corresponding’ columns have the same type.

  • For which, if any, is duplicate elimination required?

Union l.jpg



Set difference l.jpg
Set Difference


S2 – S1


Cartesian product l.jpg

  • S1  R1: Each row of S1 paired with each row of R1.

    Like the c.p for mathematical relations: every tuple of S1 “appended” to every tuple of R1

  • Q: How many rows in the result?

  • Result schemahas one field per field of S1 and R1, with field names `inherited’ if possible.

    • May have a naming conflict: Both S1 and R1 have a field with the same name.

    • In this case, can use the renaming operator…

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Cartesian Product Example



R1 X S1 =

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Rename ( )

  • Allows us to refer to a relation by more than one name and to rename conflicting names


    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.

    Ex. temp1(sid1,sname,rating, age, sid2, bid, day)(R1 x S1)

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Compound Operator: Intersection

  • In addition to the 6 basic operators, there are several additional “Compound Operators”

    • These add no computational power to the language, but are useful shorthands.

    • Can be expressed solely with the basic ops.

  • Intersection takes two input relations, which must be union-compatible.

  • Q: How to express it using basic operators?

    R  S = R  (R  S)

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Compound Operator: Join

  • Joins are compound operators involving cross product, selection, and (sometimes) projection.

  • Most common type of join is a “natural join” (often just called “join”). R S conceptually is:

    • Compute R  S

    • Select rows where attributes that appear in both relations have equal values

    • Project all unique atttributes and one copy of each of the common ones.

  • Note: Usually done much more efficiently than this.

  • Useful for putting “normalized” relations back together.

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Natural Join Example



R1 S1 =

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Other Types of Joins

  • Condition Join (or “theta-join”):

  • Result schemasame as that of cross-product.

  • May have fewer tuples than cross-product.

  • Equi-join: special case: condition c contains only conjunction of equalities.

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Compound Operator: Division

  • Useful for expressing “for all” queries like: Find sids of sailors who have reserved allboats.

  • For A/B attributes of B are subset of attrs of A.

    • May need to “project” to make this happen.

  • E.g., let A have 2 fields, x and y; B have only field y:

    A/B contains all tuples (x) such that for every y tuple in B, there is an xy tuple in A.

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Examples of Division A/B








Expressing a b using basic operators l.jpg

A/B =

Disqualified x values

Expressing A/B Using Basic Operators

  • Division is not essential op; just a useful shorthand.

    • (Also true of joins, but joins are so common that systems implement joins specially.)

  • Idea: For A/B, compute all x values that are not `disqualified’ by some y value in B.

    • x value is disqualified if by attaching y value from B, we obtain an xy tuple that is not in A.

  • Disqualified x values =

Banking example l.jpg
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)

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Example Queries

  • Find all loans of over $1200

    amount >1200 (loan)

  • Find the loan number for each loan of an amount greater than $1200

    ploan-number (amount> 1200 (loan))

loan (loan-number, branch-name, amount)

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Example Queries

  • Find the names of all customers who have a loan, an depositor account, or both, from the bank

    pcustomer-name (borrower)  pcustomer-name (depositor)

  • Find the names of all customers who have a loan and a depositor account at bank.

    pcustomer-name (borrower)  pcustomer-name (depositor)

depositor (customer-name, account-number)

borrower (customer-name, loan-number)

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Example Queries

  • Find the names of all customers who have a loan at the Perryridge branch but do not have an depositor account at any branch of the bank.

    pcustomer-name (branch-name = “Perryridge” (borrower loan))

loan (loan-number, branch-name, amount)

depositor (customer-name, account-number)

borrower (customer-name, loan-number)

– pcustomer-name(depositor)

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Example Queries

account (account-number, branch-name, balance)

Find the largest account balance

  • Rename account relation as d

  • The query is:

p balance(account) -


account.balance < d.balance(account x rd (account)))

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Example Queries

account (account-number, branch-name, balance)

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

    • Query 1

      pCN(BN=“Downtown”(depositoraccount)) 


      where CN denotes customer-name and BN denotes branch-name.

    • Query 2

      customer-name, branch-name(depositoraccount)  temp(branch-name) ({(“Downtown”), (“Uptown”)})

depositor (customer-name, account-number)

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Example Queries

  • Find all customers who have an account at all branches located in Boston.pcustomer-name, branch-name(depositoraccount)  pbranch-name(branch-city = “Boston”(branch))

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Extended Relational Operations

  • Additional Operators that extend the power of the language

    • Based on SQL… make the language less clean

  • Generalized projections

  • Outer Joins

  • Update

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General Projection


pe1, e2, …, en (Relation)

ei: can include any arithmetic operation – not only attributes

credit =



pcname, limit – balance =

Outer joins l.jpg
Outer Joins




loan borrower =

  • Join result loses:

    • any record of Perry

    • any record of Smith

Outer join l.jpg
Outer Join ( )

  • Left outer Join ( )

    preserves all tuples in left relation

loan borrower =

  • Right outer Join ( )

    preserves all tuples in right relation

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Outer Join (cont)

  • Full Outer Join ( )

    • preserves all tuples in both relations

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Update ()

  • Deletion: r  r – s

    • account  account – s bname = Perry (account)

  • Insertion: r  r s

    • branch  branch {( BU, Boston, 9M)}

  • Update: r  pe1, e2, …, en (r)

    • account  pbname, acct_no, bal * 1.05 (account)