TheRela(onalDataModeland

Rela(onalAlgebra

Debapriyo Majumdar DBMS – Fall 2016

Indian Statistical Institute Kolkata

TheRela(onalModel§  Codd proposed the relational model in 1970 §  Simple and elegant §  A relation is a table with rows and columns §  A database is a collection of one or more relations §  Advantages –  Easy to understand –  Permits the use of high level language to query the data –  Simplicity

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ARela(on:example

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A9ributes/Columns/Fields

Tuples/Records/Rows

Fieldnames

A9ributetypes§  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§  The special value null is a member of every domain.

Indicated that the value is “unknown”§  The null value causes complications in the definition of

many operations

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Rela(on:Formally§  Let A1, A2, …, An be attributes, taking values in sets D1, D2,

…., Dn respectively §  A relation schema R = (A1, A2, …, An ) is a subset of D1 × D2 × … × Dn

§  In other words, a relation r(R) is a set of n-tuples (a1, a2, …, an) where each ai ∈ Di

§  Example: Employee (emp_no, birth_date, first_name, last_name, gender, hire_date )

§  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 §  Relations are unordered; order of the tuples does not matter

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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)

–  Example: {emp_no} and {emp_no, first_name} are both superkeys of employees

§  Superkey K is a candidate key if K is minimal Example: {emp_no} is a candidate key for employees

§  One of the candidate keys is selected to be the primary key

–  Which one? A design question. §  Foreign key constraint: Value in one relation must appear in another

–  Referencing relation –  Referenced relation

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Rela(onalQueryLanguage§  “Pure” languages:–  Relational algebra–  Tuple relational calculus–  Domain relational calculus

§  The above 3 pure languages are equivalent in computing power

§  Relational Algebra–  Not turning-machine equivalent–  consists of 6 basic operations

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Select–Selec(onofrows

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§  Schema R(A, B, C, D) §  Relation r(R)

§  σ(A=B) ∧ (D > 5) (r)

A B C D a a 1 7 a b 5 7 b b 12 3 b b 23 10

A B C D a a 1 7 b b 23 10

Project–Selec(onofcolumns

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§  Schema R(A, B, C, D) §  Relation r(R)

§  ∏A,C (r)

A B C D a a 1 7 a b 5 7 b b 12 3 b b 23 10

A C a 1 a 5 b 12 b 23

Unionoftworela(ons

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§  Schema R(A, B) §  Relation r(R) and s(R) A B

a 1 b 12 b 23

A B a 1 a 5 b 23

r s

r∪ s A B a 1 b 12 b 23 a 5

Setdifferenceoftworela(ons

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§  Schema R(A, B) §  Relation r(R) and s(R) A B

a 1 b 12 b 23

A B a 1 a 5 b 23

r s

r − s A B b 12

Setintersec(onoftworela(ons

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§  Schema R(A, B) §  Relation r(R) and s(R) A B

a 1 b 12 b 23

A B a 1 a 5 b 23

r s

r − s A B a 1 b 23

Joiningtworela(ons:Cartesianproduct

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§  Schema R(A, B) and S(C, D)

§  Relation r(R) and s(S) A B a 1 b 12 b 23

C D b 10 c 5

r s

r× sA B C D a 1 b 10 b 12 b 10 b 23 b 10 a 1 c 5 b 12 c 5 b 23 c 5

Cartesianproduct:Namingofcolumns

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§  Schema R(A, B) and S(B, D)

§  Relation r(R) and s(S) A B a 1 b 12 b 23

B D b 10 c 5

r s

r× sA r.B s.B D a 1 b 10 b 12 b 10 b 23 b 10 a 1 c 5 b 12 c 5 b 23 c 5

Denote columns

prefixed by the relation they come

from

Renamingarela(on(table)§  Allows us to refer to a relation r by more than one name ρx(r) returns the expression r under the name x

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A B b 10 c 5

r A B b 10 c 5

x = ρx(r)

r.A r.B x.A x.B b 10 b 10 c 5 b 10 b 10 c 5 c 5 c 5

r×ρx (r)

Composi(onofopera(ons§  Expressions using multiple operations

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A B C D a 1 b 10 b 12 b 10 b 23 b 10 a 1 c 5 b 12 c 5 b 23 c 5

r× s

σ A=C (r× s)A B C D b 12 b 10 b 23 b 10

Naturaljoin§  Let r and s be relations on schemas R and S respectively §  Then, the natural join of relations R and 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

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Naturaljoin:example

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A B C D a a 1 7 a b 5 7 b b 12 3 b b 23 10

B D E a 7 3 a 5 2 b 10 1

s r

r s A B C D E a a 1 7 3 b b 23 10 1

Referencesandacknowledgements §  Database Management Systems: book by Raghu

Ramakrishnan and Johannes Gehrke, 3rd Edition –  Supporting teaching material at:

§  Database System Concepts: book by Silberschatz, Korth and Sudarshan –  Some of the slides are adapted from the slides at:

www.db-book.com

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