11266 Ch6 the Relational Calculus

8/13/2019 11266 Ch6 the Relational Calculus

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The Relational Calculus


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High-level, non-procedural language

Relational calculus expression declares the requirementstuples in the resulting relation must satisfy.

Provides foundation for standard query language (SQL)

Expressive power

Any expression in relational algebra can be expressed inrelational calculus.

The tuple relational calculus

The domain relational calculus


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The Tuple Relational Calculus

A tuple variable

Ranges over a database relation, denoted R(t)

Reference to an attribute of a tuple is denoted using the. notation

Tuple relational expression

Form: {t | COND(t)}, where COND(t) is a conditional

expression involving t


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The Tuple Relational Calculus

Example:

{t.NAME | EMPLOYEE(t) ANDt.SALARY>50000}

The range relation of t is EMPLOYEE The select condition that must evaluate to true:

SALARY>50000

The requested attribute: NAME


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Expressions and Formulas

A general expression

{t1.Aj, t2.Ak,, tn.Ap| COND(t1, t2,, tn, tn+1,, tn+m)}

t1, t2,, tn, tn+1,, tn+m: tuple variables ti.Aj: attribute Ajof the relation on which tiranges

COND is a condition or formula made of predicate

calculus atoms


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Expressions and Formulae

Atom

R(t): R is a relation name and t is a tuple variable

ti.A opt

j.B: A and B are attributes of the relations on

which tiand tjrange, respectively; op{=, , , }

ti.A opc: c is a constant value in the domain of A

Formula

An atom is a formula.

If F is a formula, then so is NOT(F).

If F1and F2are formulas, then so are (F1ANDF2), (F1ORF2), and (IF F1THENF2).


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Quantifiers

Existential quantifier If F is a formula, then so is (t)(F), where t is a tuple

variable.

(t)(F) is true if there exists some tuple that makes F

true.

read as "there exists an x such that...".

Universal quantifier

If F is a formula, then so is (

t)(F), where t is a tuplevariable.

(t)(F) is true if F is true for every substitution for t.

read as "for all x...".


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Other Notations

R(t) tR

NOTF F

F1ANDF2 F1

F2F1ORF2 F1F2

IFF1THENF2 F1F2


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Examples

{t.FNAME,t.LNAME | EMPLOYEE(t) and t.SALARY >50000 }

is equivalent to

SELECT T.FNAME, T.LNAME

FROM EMPLOYEE TWHERE T.SALARY > 50000


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Find all the tuples in student relation.

{t|t student}

Find those tuples for which age is greater

than 18.

{t|t student [age]>18 }

Find the roll numbers of all the students.

{t| s student (t[rollno]=s{rollno})}


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The Domain Relational Calculus

Variables

Range over single values from domains of attributes

Domain relational expression {x1, x2,,xn| COND(x1, x2,,xn, xn+1, xn+2,,xn+m)}

x1, x2,,xn, xn+1, xn+2,,xn+mare domain variables

COND is a condition or formula of the domain

relational calculus,


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The Domain Relational Calculus

Atom

R(x1, x2,,xj): R is a relation of degree j and xiis adomain variable.

xiopxj: xiand xjare domain variables and op{=, , , }

xiopc: c is a constant value in the domain of xi

Formula

Made of atoms, variables, and quantifiers


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Examples

Find the loan-no, branch-name and amount for loans of over

20000.

{| loan a>2000}

Find all the loan numbers for loans with amount greater than

20000.

{| b,a( loan a >20000)}

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11266 Ch6 the Relational Calculus