Rensselaer Polytechnic Institute
CSCI-4380 – Database Systems
David Goldschmidt, Ph.D.

A functional dependency on relation R is a
logical expression of the form X  Y
 X and Y are sets of attributes of R
▪ i.e. X = { A1, A2, ..., An } and Y = { B1, B2, ..., Bm }
where n = m or n <> m
 X  Y means that whenever any pair of tuples
in R have the same values for attributes in X,
then they must also have the same values for
attributes in Y

For each X  Y defined on relation R,
it means that X functionally determines Y
 Or more specifically, the attributes of X
functionally determine the attributes of Y

More generally, a functional dependency
adds meaning to attributes of R
 In some cases, the occurrence of duplicate
tuples does not make semantic sense

For a given relation R, we look at the set of
all functional dependencies to tell us what
tuples we can (and should) store

We can also reason by applying simple
inference rules to the tuples
 e.g. transitivity, splitting/combining, etc.

A constraint of any kind on relation R is said
to be trivial if it holds for every instance of R
 If Y  X, then X  Y is true for all relations
 In other words, a trivial functional dependency
has a right-hand side (Y) that is a subset of its
left-hand side (X)
 e.g. name artist  name
 e.g. name  name

The functional dependency X  Y is
equivalent to X  Z where attributes
of Z are all those attributes of Y that
are not also attributes of X
 In other words, some of the attributes on the
right-hand side (of X  Y) are also on the left (X)
 We can simplify this by removing attributes from
the right-hand side that also appear on the left

Given functional dependency
 XY

We can always add a set Z of attribute(s)
 XZ  YZ

This is called augmentation

Given functional dependency X  Y as
 A1, A2, ..., An  B1, B2, ..., Bm

We can split it into multiple functional
dependencies (singletons) as follows:
 A1, A2, ..., An  B1
 A1, A2, ..., An  B2
 ...
 A1, A2, ..., An  Bm

Given functional dependencies as follows:
 A1, A2, ..., An  B1
 A1, A2, ..., An  B2
 ...
 A1, A2, ..., An  Bm

We can combine attributes on the righthand side to form functional dependency
 A1, A2, ..., An  B1, B2, ..., Bm

Given functional dependencies
 X  Y and Y  Z

We can unequivocally conclude that
 XZ

And if some attributes of Z are also
attributes of X, we can eliminate them from
the right-hand side (trivial-dependency rule)

For a given relation R, we look at the set of
functional dependencies to identify which
attribute(s) imply all the rest
 These attribute(s) form a key on R

Set K = { A1, A2, ..., An } is a key on R if:
 K functionally determine all other attributes of R
 No proper subset of K functionally determines
all other attributes of R

By definition, a key must be unique

A key K must functionally determine all
other attributes of relation R
 e.g. Student( id, name, address )
 The key is the id attribute

By definition, a key must be minimal

No proper subset of key K can functionally
determine all other attributes of relation R
 e.g. Student( id, name, address )
 Even though id and name together might be
unique, the id attribute is minimal

A set of attributes that contains a key
is called a superkey (a superset of a key)
 The uniqueness constraint must be satisfied
 The minimality constraint need not be satisfied
 Every key is a superkey
 e.g. Student( id, name, address )
 Attribute id is both a key and a superkey
 Attributes (id, name) form a superkey

Model a US Census relation
 Name, SSN, address, city, state, zip,
area code, phone number, etc.
 Use only a single relation
 Describe functional dependencies
 Identify keys and superkeys

Given relation R with attributes A, B, C, D, E
and A  BC, CD  E, BE  C
 What does AE functionally determine (infer)?
 In other words, AE  _____?

Given a set of attributes X, the
closure X+ is the set of attributes
functionally determined by X

Given a relation R and a set F of functional
dependencies, we need a way to find
whether a functional dependency X  Y is
true with respect to F

Given relation R with attributes A, B, C, D, E
and A  BC, CD  E, BE  C
 AE  _____?
 From reflexivity, AE+ = { A, E }
 From A  BC, AE+ = { A, B, C, E }
 No other rules are applicable or add to AE+
 We conclude that AE  ABCE or simply AE  BC
 Or AE  A, AE  B, AE  C, and AE  E

Given a set F of functional dependencies,
the closure X+ of a set of attributes X is
determined by the following algorithm:
 Initialize X+ to X
 Repeat until X+ does not change:
▪ Find any unapplied functional dependency Y  Z
in F such that Y  X+
▪ Set X+ = X+  Z

A set F of functional dependencies implies a
functional dependency X  Y if Y  X+

In other words, if Y is in the closure of X,
then functional dependency X  Y is true

A key K for a given relation R is a minimal set
of attributes A1, A2, ..., An such that closure
{A1, A2, ..., An}+ is the set of all attributes of R
 MusicGroup(name, artist, genre, dateformed,
datefirstjoined)
 name  genre dateformed
 name artist  datefirstjoined
 K must be (name, artist) because K+ = {name,
artist, genre, dateformed, datefirstjoined}

Review the US Census relation
 What other functional dependencies
can you infer?
 Pick pairs of attributes (e.g. name and
state) and identify the resulting closure
 In other words, what is the set of attributes X+
functionally determined by set X (the pair)?