Chapter 3: Relational Model Structure of Relational Databases - PDF document

' $ Chapter 3: Relational Model • Structure of Relational Databases • Relational Algebra • Tuple Relational Calculus • Domain Relational Calculus • Extended Relational-Algebra-Operations • Modification of the Database • Views & % Database Systems Concepts 3.1 Silberschatz, Korth and Sudarshan c � 1997 ' $ Basic Structure • Given sets A 1 , A 2 , ..., A n a relation r is a subset of A 1 × A 2 × ... × A n Thus a relation is a set of n-tuples ( a 1 , a 2 , ..., a n ) where a i ∈ A i • Example: If customer-name = { Jones, Smith, Curry, Lindsay } customer-street = { Main, North, Park } customer-city = { Harrison, Rye, Pittsfield } Then r = { (Jones, Main, Harrison), (Smith, North, Rye), (Curry, North, Rye), (Lindsay, Park, Pittsfield) } is a relation over customer-name × customer-street × customer-city & % Database Systems Concepts 3.2 Silberschatz, Korth and Sudarshan c � 1997

' $ Relation Schema • A 1 , A 2 , ..., A n are attributes • R = ( A 1 , A 2 , ..., A n ) is a relation schema Customer-schema = ( customer-name, customer-street, customer-city ) • r ( R ) is a relation on the relation schema R customer ( Customer-schema ) & % Database Systems Concepts 3.3 Silberschatz, Korth and Sudarshan c � 1997 ' $ Relation Instance • 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. customer-name customer-street customer-city Jones Main Harrison Smith North Rye Curry North Rye Lindsay Park Pittsfield customer & % Database Systems Concepts 3.4 Silberschatz, Korth and Sudarshan c � 1997

' $ 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 ). By “possible r ” we mean a relation r that could exist in the enterprise we are modeling. Example: { customer-name, customer-street } and { customer-name } are both superkeys of Customer , if no two customers can possibly have the same name. • K is a candidate key if K is minimal Example: { customer-name } is a candidate key for Customer , since it is a superkey (assuming no two customers can possibly have the same name), and no subset of it is a superkey. & % Database Systems Concepts 3.5 Silberschatz, Korth and Sudarshan c � 1997 ' $ Determining Keys from E-R Sets • Strong entity set. The primary key of the entity set becomes the primary key of the relation. • Weak entity set. 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. • Relationship set. The union of the primary keys of the related entity sets becomes a super key of the relation. For binary many-to-many relationship sets, above super key is also the primary key. For binary many-to-one relationship sets, the primary key of the “many” entity set becomes the relation’s primary key. For one-to-one relationship sets, the relation’s primary key can be that of either entity set. & % Database Systems Concepts 3.6 Silberschatz, Korth and Sudarshan c � 1997

' $ Query Languages • Language in which user requests information from the database. • Categories of languages: – Procedural – Non-procedural • “Pure” languages: – Relational Algebra – Tuple Relational Calculus – Domain Relational Calculus • Pure languages form underlying basis of query languages that people use. & % Database Systems Concepts 3.7 Silberschatz, Korth and Sudarshan c � 1997 ' $ Relational Algebra • Procedural language • Six basic operators – select – project – union – set difference – Cartesian product – rename • The operators take two or more relations as inputs and give a new relation as a result. & % Database Systems Concepts 3.8 Silberschatz, Korth and Sudarshan c � 1997

' $ Select Operation • Notation: σ P ( r ) • Defined as: σ P ( r ) = { t | t ∈ r and P ( t ) } Where P is a formula in propositional calculus, dealing with terms of the form: < attribute > = < attribute > or < constant > � = > ≥ < ≤ “connected by”: ∧ ( and ), ∨ ( or ), ¬ ( not ) & % Database Systems Concepts 3.9 Silberschatz, Korth and Sudarshan c � 1997 ' $ Select Operation – Example • Relation r : A B C D α α 1 7 α β 5 7 β β 12 3 β β 23 10 • σ A = B ∧ D > 5 ( r ) A B C D 1 7 α α 23 10 β β & % Database Systems Concepts 3.10 Silberschatz, Korth and Sudarshan c � 1997

' $ Project Operation • Notation: Π A 1 , A 2 , ..., A k ( r ) 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 & % Database Systems Concepts 3.11 Silberschatz, Korth and Sudarshan c � 1997 ' $ Project Operation – Example • Relation r : A B C α 10 1 α 20 1 β 30 1 β 40 2 • Π A, C ( r ) A C A C 1 1 α α — — 1 = 1 α β β 1 β 2 β 2 & % Database Systems Concepts 3.12 Silberschatz, Korth and Sudarshan c � 1997

' $ 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., 2nd column of r deals with the same type of values as does the 2nd column of s ) & % Database Systems Concepts 3.13 Silberschatz, Korth and Sudarshan c � 1997 ' $ Union Operation – Example • Relations r , s : A B A B α 1 α 2 2 3 α β 1 s β r • r ∪ s A B α 1 2 α 1 β 3 β & % Database Systems Concepts 3.14 Silberschatz, Korth and Sudarshan c � 1997

' $ 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 & % Database Systems Concepts 3.15 Silberschatz, Korth and Sudarshan c � 1997 ' $ Set Difference Operation – Example • Relations r , s : A B A B 1 2 α α 2 3 α β 1 s β r • r − s A B 1 α 1 β & % Database Systems Concepts 3.16 Silberschatz, Korth and Sudarshan c � 1997

' $ Cartesian-Product Operation • Notation: r × s • Defined as: r × 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 of r ( R ) and s ( S ) are not disjoint, then renaming must be used. & % Database Systems Concepts 3.17 Silberschatz, Korth and Sudarshan c � 1997 ' $ Cartesian-Product Operation – Example • Relations r , s : A B C D E α 1 α 10 + 2 10 + β β r 20 − β γ 10 − s • r × s A B C D E 1 10 + α α α 1 β 10 + 1 20 − α β 1 10 − α γ 2 10 + β α 2 10 + β β β 2 β 20 − 2 10 − β γ & % Database Systems Concepts 3.18 Silberschatz, Korth and Sudarshan c � 1997

' $ Composition of Operations • Can build expressions using multiple operations • Example: σ A = C ( r × s ) • r × s – Notation: r 1 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 t r from r and t s from s . – If t r and t s 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 t r on r ∗ t has the same value as t s on s & % Database Systems Concepts 3.19 Silberschatz, Korth and Sudarshan c � 1997 ' $ Composition of Operations (Cont.) Example: R = ( A, B, C, D ) S = ( E, B, D ) • Result schema = ( A, B, C, D, E ) • r 1 s is defined as: Π r.A,r.B,r.C,r.D,s.E ( σ r.B = s.B ∧ r.D = s.D ( r × s )) & % Database Systems Concepts 3.20 Silberschatz, Korth and Sudarshan c � 1997

' $ Natural Join Operation – Example • Relations r , s : A B C D B D E 1 a 1 a α α α 2 a 3 a β γ β γ 4 β b 1 a γ α 1 γ a 2 b δ 2 b 3 b δ β ǫ r s • r 1 s A B C D E 1 a α α α 1 a α α γ α 1 γ a α α 1 γ a γ & % 2 b δ β δ Database Systems Concepts 3.21 Silberschatz, Korth and Sudarshan c � 1997 ' $ Division Operation r ÷ s • 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 , ..., A m , B 1 , ..., B n ) – S = ( B 1 , ..., B n ) The result of r ÷ s is a relation on schema R − S = ( A 1 , ..., A m ) r ÷ s = { t | t ∈ Π R − S ( r ) ∧ ∀ u ∈ s ( tu ∈ r ) } & % Database Systems Concepts 3.22 Silberschatz, Korth and Sudarshan c � 1997