Chapter 11
Functional Dependencies
Prof. Yin-Fu Huang
CSIE, NYUST
11.1 Introduction
 A functional dependency (FD) is basically a many-toone relationship from one set of attributes to another
within a given relvar.
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11.2 Basic Definitions
 To distinguish between (a) the value of a given relvar at a
given point in time and (b) the set of all possible values that
the given relvar might assume at different times
Case (a):
 Let r be a relation, and let X and Y be arbitrary subsets of
the set of attributes of r. Y is functionally dependent on X,
X →Y, if and only if each X value in r has associated with
it precisely one Y value in r.
 Sample value (See Fig. 11.1)
{S#}→{City}
{S#,P#}→{S#,P#,City,Qty}
{S#}→{Qty}
{Qty}→{S#}
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11.2 Basic Definitions (Cont.)
Case (b):
 Let R be a relation variable, and let X and Y be arbitrary subsets
of the set of attributes of R. Y is functionally dependent on X,
X →Y, if and only if in every possible legal value of R, each X
value has associated with it precisely one Y value.
{S#}→{Qty}
and
{Qty}→{S#} do not hold “for all time”
 If relvar R satisfies the FD A →B and A is not a candidate key, then
R will necessarily involve some redundancy.
 To find some way of reducing the set of FDs to a manageable size.
 The reason is that FDs represent certain integrity constraints, and we
would thus like the DBMS to enforce them.
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11.3 Trivial and Nontrivial Dependencies
 An FD is trivial if and only if the right-hand side is a
subset of the left-hand side.
{S#,P#}→{S#}
 The trivial dependencies can be eliminated.
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11.4 Closure of a Set of Dependencies
 The set of all FDs that are implied by a given set S of FDs is
called the closure of S, written S+ .
 Armstrong‘s axioms:
1. Reflexivity: If B is a subset of A, A →B.
2. Augmentation: If A →B, then AC →BC.
3. Transitivity: If A →B and B →C, then A →C.
 The rules are complete and sound.
4. Self-determination: A →A.
5. Decomposition: If A →BC, then A →B and A →C.
6. Union: If A →B and A →C, then A →BC.
7. Composition: If A →B and C →D, then AC →BD.
8. General Unification Theorem: If A →B and C →D,
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A ∪(C-B)→BD.
11.4 Closure of a Set of Dependencies (Cont.)
 Example: R:{A,B,C,D,E,F} and the FDs
A →BC
B →E
CD →EF
AD →F is a member of the closure of the given set.
1. A →BC (given)
2. A →C (1, decomposition)
3. AD →CD (2, augmentation)
4. CD →EF (given)
5. AD →EF (3 and 4, transitivity)
6. AD →F (5, decomposition)
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11.5 Closure of a Set of Attributes
 Given a relvar R, a set Z of attributes of R, and a set S of FDs
that hold for R, we can determine the set of attributes of R that
are functionally dependent on Z-the closure Z+ of Z under S.
 A simple algorithm for computing the closure Z+
(See Fig. 11.2)
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11.5 Closure of a Set of Attributes (Cont.)
 Example: R:{A,B,C,D,E,F} and the FDs
A →BC
E →CF
B →E
CD →EF




Computing the closure {A,B} + ={A,B,C,E,F}
Given a set S of FDs, we can easily tell whether a specific FD
X →Y follows from S, because that FD will follow if and only
if Y is a subset of the closure X+ of X under S.
A superkey for a relvar R is a set of attributes of R that
includes some candidate key of R as a subset.
K is a superkey if and only if the closure K+ of K under the
given set of FDs is precisely the set of all attributes of R.
K is a candidate key Advanced
if and only
if itSystem
is an irreducible superkey.
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11.6 Irreducible Sets of Dependencies (1/3)
 If every FD implied by S1 is implied by S2, S2 is a cover for S1.
 Equivalence
 A set S of FDs to be irreducible if and only if it satisfies the
following three properties:
1. The right side of every FD in S involves just one
attribute.
2. The left side of every FD in S is irreducible, meaning
that no attribute can be discarded from the determinant
without changing the closure S+ .
3. No FD in S can be discarded from S without changing the
closure S+ .
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11.6 Irreducible Sets of Dependencies (2/3)
 Example:
1. P#→{Pname,Color}
P#→Weight
P#→City
2. {P#,Pname}→Color
P#→Pname
P#→Weight
P#→City
3. P#→P#
P#→Pname
P#→Color
P#→Weight
P#→City
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11.6 Irreducible Sets of Dependencies (3/3)
 For every set of FDs, there exists at least one equivalent set
that is irreducible.
 Example: R:{A,B,C,D} and the FDs
A →BC
B →C
A →B
AB →C

A →B
B →C
A →D
AC →D
 A given set of FDs does not necessarily have a unique
irreducible equivalent.
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The End.
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