Branch and Staff (part) Relations
Relational Model
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Examples of Attribute Domains
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Relational Model Terminology
o A relation is a table with columns and rows.
Only applies to logical structure of the
database, not the physical structure.
o Attribute is a named column of a relation.
o Domain is a set of allowable values for one
or more attributes.
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CS3462 Introduction to Database Systems
Helena Wong, 2001
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Relational Model Terminology
Alternative Terminology (Relational Model)
o Tuple is a row of a relation.
o Degree is a number of attributes in a
relation.
o Cardinality is a number of tuples in a
relation.
o Relational Database is a collection of
normalized relations.
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Mathematical Relations
Mathematical Relation
Mathematical definition of relation
– Consider two sets, D1 and D2, where D1 = {2, 4}
and D2 = {1, 3, 5}.
– Cartesian product is D1 × D2, the set of all
ordered pairs, first element is member of D1
and second element is member of D2.
– Alternative, find all combinations of elements
with first from D1 and second from D2.
D1 ×=D2 = {(2, 1), (2, 3), (2, 5), (4, 1), (4, 3), (4, 5)}
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CS3462 Introduction to Database Systems
Helena Wong, 2001
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Any subset of Cartesian product is a relation. Eg.,
– R = {(2, 1), (4, 1)}
– Using same sets, form another relation, S,
where first element is always twice the
second.
S = {(x, y) | x ∈D1, y ∈D2, and x = 2y}
– Only one ordered pair in the Cartesian
Product satisfies this condition.
S = {(2, 1)}
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Mathematical Relation
Mathematical Relation
o Consider three sets D1, D2, and D3 with
Cartesian Product D1 × D2 × D3.
o To define a general relation on n domains, let
D1, D2, . . ., Dn be n sets with Cartesian
product defined as
D1 × D2 ×=. . . ×=Dn = {(d1, d2, . . . , dn) | d1 ∈D1, d2
∈D2, . . . , dn∈Dn}
D1 = {1, 3}
D2 = {2, 4}
D3 = {5, 6}
D1 × D2 × D3 = {(1,2,5), (1,2,6), (1,4,5), (1,4,6),
(3,2,5), (3,2,6), (3,4,5), (3,4,6)}
usually written as
n
XD
i
o Any subset of these ordered triples is a
relation.
i=1
o In defining relations we specify the sets, or
domains, from which we chose values.
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Properties of Relations
Relational Keys
o Relation name is distinct from all other relations.
o Superkey (Key)
– An attribute or a set of attributes that uniquely
identifies a tuple within a relation.
o Each cell of relation contains exactly one atomic
(single) value.
o Candidate Key
o Each attribute has a distinct name.
o Values of an attribute are all from the same domain.
– A superkey (K) such that no proper subset is a
superkey within the relation.
o Order of attributes has no significance.
– In each tuple of R, the values of K uniquely identify
that tuple (uniqueness).
o Each tuple is distinct; there are no duplicate tuples.
– No proper subset of K has the uniqueness property
(irreducicility).
o Order of tuples has no significance, theoretically.
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CS3462 Introduction to Database Systems
Helena Wong, 2001
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Relational Keys
Relational Integrity
o Primary Key
– Candidate key selected to identify tuples uniquely
within relation.
o Null
o Alternate Keys
– Candidate keys that are not selected to be the
primary key.
o Foreign Key
– An attribute or set of attributes within one relation that
matches candidate key of some (possibly same)
relation.
– Represents a value for an attribute that is
currently unknown or is not applicable for this
tuple.
– Deals with incomplete or exceptional data.
– Null represents the absence of a value and is
not the same as zero or spaces, which are
values.
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Relational Integrity
Relational Integrity
2 principal rules for the relational model:
Enterprise Constraints
o Entity Integrity
– In a base relation, no attribute of a primary key can
be null.
o Referential Integrity
– Additional rules specified by users or
database administrators.
– Eg., the maximum number of staff in a
branch.
– If foreign key exists in a relation, either the foreign
key value must match a candidate key value of some
tuple in its home relation or foreign key value must be
wholly null.
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CS3462 Introduction to Database Systems
Helena Wong, 2001
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Relational Algebra
Relational Algebra
o Relational algebra operations work on one or more
relations to define another relation without changing
the original relations.
o 5 basic operations in relational algebra:
Selection, Projection, Cartesian Product, Union,
and Set Difference.
o Thus, both operands and results are relations, so
output from one operation can become input to
another operation.
o Performs most of the data retrieval operations
needed.
o This allows expressions to be nested, just as in
arithmetic. This property is called closure.
o Also have Join, Intersection, and Division
operations that can be expressed in terms of 5 basic
operations.
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Relational Algebra Operations
Relational Algebra Operations
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CS3462 Introduction to Database Systems
Helena Wong, 2001
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Selection
Example - Selection
Selection Operation
o List all staff with a salary greater than
£10,000: σsalary > 10000 (Staff)
σpredicate (R)
works on a single relation R and defines a
relation that contains only those tuples
(rows) of R that satisfy the specified
condition (predicate).
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Projection
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Example - Projection
o Produce a list of salaries for all staff, showing only the
Sno, FName, LName, and Salary details:
Projection Operation
Π sno, fname, lname, salary(Staff)
Πcol1, . . . , coln(R)
works on a single relation R and defines a
relation that contains a vertical subset of R,
extracting the values of specified attributes
and eliminating duplicates.
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CS3462 Introduction to Database Systems
Helena Wong, 2001
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Cartesian Product
Example - Cartesian Product
o List the names and comments of all renters who have
viewed a property.
Π rno, pno,comment (Viewing))
(Π
Π rno, fname, lname(Renter)) X (Π
Cartesian Product
RXS
defines a relation that is the concatenation
of every tuple of relation R with every tuple
of relation S.
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Eg. Cartesian Product and Selection
Union
o Use selection operation to extract those tuples
where Renter.Rno = Viewing.Rno.
∏rno,fname,lname(Renter)) Χ
σrenter.rno = viewing.rno((∏
(∏
∏rno,pno,comment(Viewing)))
Union Operation
o Cartesian product and Selection can be reduced to a
single operation called a join.
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CS3462 Introduction to Database Systems
Helena Wong, 2001
R∪S
Union of two relations R and S with I and J
tuples, respectively, is obtained by
concatenating them into one relation with a
maximum of (I + J) tuples, duplicate tuples
being eliminated. R and S must be unioncompatible.
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Example - Union
Set Difference
Set Difference Operation
R–S
o Construct a list of all areas where
there is either a branch or a property.
defines a relation consisting of the tuples
that are in relation R, but not in S. R and S
must be union-compatible.
Π area(Branch) ∪ Πarea (Property_for_Rent)
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Example - Set Difference
Join Operations
o Construct a list of all cities where there is a
branch office but no properties.
o Join is a derivative of Cartesian product.
Π city (Branch) – Πcity (Property_for_Rent)
o Equivalent to performing a selection, using the
join predicate as the selection formula, over the
Cartesian product of the two operand relations.
o One of the most difficult operations to
implement efficiently in a relational DBMS and
one of the reasons why relational systems have
intrinsic performance problems.
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CS3462 Introduction to Database Systems
Helena Wong, 2001
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Join Operations
Theta-join (θ-join)
o There are various forms of join operation
oR
–
Theta-join
–
Equi-join (a particular type of theta-join)
–
Natural join
–
Outer join
–
Semi-join
–
–
FS
Defines a relation that contains tuples
satisfying the predicate F from the
Cartesian product of R and S.
The predicate F is of the form R.ai θ S.bi
where θ may be one of the comparison
operators (<, < =, >, > =, =, ~ =).
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Theta-join (θ-join)
Example - Equi-join
o We can rewrite the theta-join in terms of the
basic Selection and Cartesian product
operations.
List the names and comments of all renters who
have viewed a property.
(Π
Πrno,fname,lname (Renter))
renter.rno = viewing.rno
(Π
Πrno,pno,comment(Viewing))
R
FS
= σF(R Χ S)
o Degree of a theta-join is sum of the degrees of
the operand relations R and S. If predicate F
contains only equality (=), the term equi-join is
used.
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CS3462 Introduction to Database Systems
Helena Wong, 2001
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Natural Join
Example - Natural Join
oR
o List the names and comments of all renters
who have viewed a property.
–
S
Natural join is an equi-join of the two
relations R and S over all common
attributes x. One occurrence of each
common attribute is eliminated from the
result.
(Πrno,fname,lname (Renter))
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Outer Join
Semi-join
o Often in joining two relations, there is no
matching value in the join columns. To
display rows in the result that do not have
matching values in the join column, we use
the outer join.
oR
oR
–
–
FS
The semi-join operation defines a relation
that contains the tuples of R that
participate in the join of R with S.
S
The (left) outer join is a join in which
tuples from R that do not have matching
values in the common columns of S are
also included in the result relation.
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CS3462 Introduction to Database Systems
Helena Wong, 2001
(Πrno,pno,comment(Viewing))
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Example - Semi-join
Intersection
o List complete details of all staff who work at the
branch in Partick.
Staff
staff.bno = branch.bno and branch.area = ‘Partick’
o R∩S
–
The intersection operation consists of the
set of all tuples that are in both R and S.
–
R and S must be union-compatible.
Branch
o Expressed using basic operations
R ∩ S = R – (R – S)
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Division
Example - Division
o R÷S
– The division operation consists of the set
of tuples from R defined over the
attributes C that match the combination of
every tuple in S.
o Identify all renters who have viewed all
properties with three rooms.
(Π
Π rno,pno (Viewing)) ÷ (Π
Π pno (σ
σrooms = 3 (Property_for_Rent)))
o Expressed using basic operations
T1 = ΠC(R)
T2 = ΠC((S X T1) – R)
T = T1 – T 2
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CS3462 Introduction to Database Systems
Helena Wong, 2001
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Views
Views
o Base Relation
A named relation, corresponding to an entity
in conceptual schema, whose tuples are
physically stored in database.
o View
Dynamic result of one or more relational
operations operating on the base relations
to produce another relation.
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Views
Purpose of Views
o A view is a virtual relation that does not actually
exist in the database but is produced upon request,
at time of request.
o Provides a powerful and flexible security
mechanism by hiding parts of database from certain
users.
o Contents of a view are defined as a query on one or
more base relations.
o Permits users to access data in a customized way,
so that same data can be seen by different users in
different ways, at same time.
o Views are dynamic, meaning that changes made to
base relations that affect view attributes are
immediately reflected in the view.
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CS3462 Introduction to Database Systems
Helena Wong, 2001
o It can simplify complex operations on base
relations.
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Updating Views
Updating Views
o All updates to a base relation should be
immediately reflected in all views that
reference that base relation.
o However, there are restrictions on types of
modifications that can be made through views:
o If view is updated, underlying base relation
should reflect change.
– Updates are allowed if query involves a single
base relation and contains a candidate key of
base relation.
– Updates are not allowed involving multiple base
relations.
– Updates are not allowed involving aggregation or
grouping operations.
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Views - Advantages / Disadvantages
Advantages
Disadvantages
Data independence
Update restriction
Currency
Structure restriction
Improved security
Performance
Reduced complexity
Convenience
Customization
Data integrity
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CS3462 Introduction to Database Systems
Helena Wong, 2001
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