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Relational Algebra. Dashiell Fryer. What is Relational Algebra?. Relational algebra is a procedural query language. It consists of the select, project, union, set difference, Cartesian product, and rename operations.

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Relational Algebra


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relational algebra

Relational Algebra

Dashiell Fryer

what is relational algebra
What is Relational Algebra?
  • Relational algebra is a procedural query language.
  • It consists of the select, project, union, set difference, Cartesian product, and rename operations.
  • Set intersection, division, natural join, and assignment combine the fundamental operations.
select operation
Select Operation
  • Select operation returns a relation that satisfies the given predicate from the original relation.
  • Denoted by lowercase sigma σ.
  • Example: σbranch-name = “Perryridge”(loan) from page 89 returns a relation of the tuples from relation loan whose branch is Perryridge.
  • =,≠,≤,≥,<,>, Λ,¬,v are legal in the predicate.
project
Project
  • Unary operation
  • Denoted by uppercase pi Π.
  • Returns a relation with only the specified attributes.
  • Example: page 90 Πloan-number, amount (loan) lists all of the loan numbers and the amounts.
composition of operations
Composition of Operations
  • Results of relational operations are relations themselves.
  • Compositions of operations form a relational-algebra expression.
union
Union
  • Denoted, as in set theory, by .
  • Binary operation
  • Results in a relation with all of the tuples that appear in either or both of the argument relations.
union cont d
Union (Cont’d)
  • Unions must be between compatible relations
  • Both relations must have the same number of attributes.
  • Domains of the ith attribute of the first and the ith attribute of the second must be the same for all i.
set difference
Set Difference
  • Denoted by -
  • Binary operation
  • R – S produces all tuples in R but not in S
  • Relations must be compatible under the same conditions as the union operation.
cartesian product
Cartesian-Product
  • Denoted by a cross (×)
  • Binary operation
  • Combines any two relations
  • R × S is the Cartesian-Product of R and S
  • Output has the attributes of both relations
cartesian product cont d
Cartesian-Product (Cont’d)
  • Repeated attribute names are preceded by the relation they originated from.
  • Example: page 93 r= borrower × loan

(borrower.customer-name,

borrower.loan-number, loan.loan-number, loan.branch-name, loan.amount)

rename
Rename
  • Unary operation
  • Denoted by lowercase Greek letter rho ρ
  • Useful for naming the unnamed relations returned from other operations.
formal definition
Formal Definition
  • Relational Algebra acts on either a relation in the database or a constant relation.
  • Suppose E1 and E2 are relational-algebra expressions
formal definition cont d
Formal Definition (Cont’d)
  • All relational-algebra expressions are:

E1 E2

E1 – E2

E1 × E2

σP(E1), P is the predicate

ΠS(E1), S is a list of attributes in E1

ρx(E1), x is the new name of E1

additional operations
Additional Operations
  • The fundamental operations can be composed to make new, simpler operations.
  • Set-Intersection
  • Natural-Join
  • Division
  • Assignment
set intersection
Set-Intersection
  • Denoted by the symbol .
  • Results in a relation that contains only the tuples that appear in both relations.
  • R  S = R – (R – S)
  • Since set-intersection can be written in terms of set-difference, it is not a fundamental operation.
natural join
Natural-Join
  • Denoted by |x|.
  • Binary operation
  • Creates a Cartesian-product of the arguments then performs selection to force equality on attributes that appear in both relations
division
Division
  • Denoted by 
  • Binary Operation
  • Used in queries that include the phrase “for all”.
division cont d
Division (Cont’d)
  • Division is an operation on schema R – S
  • A tuple t is in r  s if and only if:
  • t is in ΠR – S(r) and
  • For every tuple ts in s, there is a tuple tr in r satisfying both of the following:

a. tr[S] = ts[R]

b. tr[R – S] = t

assignment
Assignment
  • Denoted by the  symbol
  • Works like assignment operations in programming languages
  • Assignment must always be made to a temporary relation
  • Does not result in display to the user
summary
Summary
  • Relational Algebra defines a set of operations that operate on relations and result in new tables.
  • Operations can be composed
  • Additional operations simplify the algebra but do not add any additional power.