Pertemuan 5_Relation Matriks_01 (17)

April 14, Applied Discrete Mathematics 1
Combining RelationsCombining Relations
Another Example:Another Example: Let X and Y be relations onLet X and Y be relations on
A = {1, 2, 3, …}.A = {1, 2, 3, …}.
X = {(a, b) | b = a + 1} “b equals a plus 1”X = {(a, b) | b = a + 1} “b equals a plus 1”
Y = {(a, b) | b = 3a} “b equals 3 times a”Y = {(a, b) | b = 3a} “b equals 3 times a”
X = {(1, 2), (2, 3), (3, 4), (4, 5), …}X = {(1, 2), (2, 3), (3, 4), (4, 5), …}
Y = {(1, 3), (2, 6), (3, 9), (4, 12), …}Y = {(1, 3), (2, 6), (3, 9), (4, 12), …}
XX°° Y = {Y = { (1, 4),(1, 4), (2, 7),(2, 7), (3, 10),(3, 10), (4, 13),(4, 13), ……}}
Y maps an element a to the element 3a, andY maps an element a to the element 3a, and
afterwards X maps 3a to 3a + 1.afterwards X maps 3a to 3a + 1.
XX°° Y = {(a,b) | b = 3a + 1}Y = {(a,b) | b = 3a + 1}

n-ary Relationsn-ary Relations
In order to study an interesting application of relations,In order to study an interesting application of relations,
namelynamely databasesdatabases, we first need to generalize the, we first need to generalize the
concept of binary relations toconcept of binary relations to n-ary relationsn-ary relations..
Definition:Definition: Let ALet A11, A, A22, …, A, …, Ann be sets. Anbe sets. An n-ary relationn-ary relation
on these sets is a subset of Aon these sets is a subset of A11××AA22××……××AAnn..
The sets AThe sets A11, A, A22, …, A, …, Ann are called theare called the domainsdomains of theof the
relation, and n is called itsrelation, and n is called its degreedegree..

n-ary Relationsn-ary Relations
Example:Example:
Let R = {(a, b, c) | a = 2bLet R = {(a, b, c) | a = 2b ∧∧ b = 2c with a, b, cb = 2c with a, b, c∈∈ZZ}}
What is the degree of R?What is the degree of R?
The degree of R is 3, so its elements are triples.The degree of R is 3, so its elements are triples.
What are its domains?What are its domains?
Its domains are all equal to the set of integers.Its domains are all equal to the set of integers.
Is (2, 4, 8) in R?Is (2, 4, 8) in R?
No.No.
Is (4, 2, 1) in R?Is (4, 2, 1) in R?
Yes.Yes.

Databases and RelationsDatabases and Relations
Let us take a look at a type of databaseLet us take a look at a type of database
representation that is based on relations, namely therepresentation that is based on relations, namely the
relational data model.relational data model.
A database consists of n-tuples calledA database consists of n-tuples called recordsrecords, which, which
are made up ofare made up of fieldsfields..
These fields are theThese fields are the entriesentries of the n-tuples.of the n-tuples.
The relational data model represents a database asThe relational data model represents a database as
an n-ary relation, that is, a set of records.an n-ary relation, that is, a set of records.

Example:Example: Consider a database of students, whoseConsider a database of students, whose
records are represented as 4-tuples with the fieldsrecords are represented as 4-tuples with the fields
Student NameStudent Name,, ID NumberID Number,, MajorMajor, and, and GPAGPA::
R = {(Ackermann, 231455, CS, 3.88),R = {(Ackermann, 231455, CS, 3.88),
(Adams, 888323, Physics, 3.45),(Adams, 888323, Physics, 3.45),
(Chou, 102147, CS, 3.79),(Chou, 102147, CS, 3.79),
(Goodfriend, 453876, Math, 3.45),(Goodfriend, 453876, Math, 3.45),
(Rao, 678543, Math, 3.90),(Rao, 678543, Math, 3.90),
(Stevens, 786576, Psych, 2.99)}(Stevens, 786576, Psych, 2.99)}
Relations that represent databases are also calledRelations that represent databases are also called
tablestables, since they are often displayed as tables., since they are often displayed as tables.

A domain of an n-ary relation is called aA domain of an n-ary relation is called a primary keyprimary key
if the n-tuples are uniquely determined by their valuesif the n-tuples are uniquely determined by their values
from this domain.from this domain.
This means that no two records have the same valueThis means that no two records have the same value
from the same primary key.from the same primary key.
In our example, which of the fieldsIn our example, which of the fields Student NameStudent Name,, IDID
NumberNumber,, MajorMajor, and, and GPAGPA are primary keys?are primary keys?
Student NameStudent Name andand ID NumberID Number are primary keys,are primary keys,
because no two students have identical values inbecause no two students have identical values in
these fields.these fields.
In a real student database, onlyIn a real student database, only ID NumberID Number would bewould be
a primary key.a primary key.

In a database, a primary key should remain one evenIn a database, a primary key should remain one even
if new records are added.if new records are added.
Therefore, we should use a primary key of theTherefore, we should use a primary key of the
intensionintension of the database, containing all the n-tuplesof the database, containing all the n-tuples
that can ever be included in our database.that can ever be included in our database.
Combinations of domainsCombinations of domains can also uniquely identifycan also uniquely identify
n-tuples in an n-ary relation.n-tuples in an n-ary relation.
When the values of aWhen the values of a set of domainsset of domains determine an n-determine an n-
tuple in a relation, thetuple in a relation, the Cartesian productCartesian product of theseof these
domains is called adomains is called a composite keycomposite key..

We can apply a variety ofWe can apply a variety of operationsoperations on n-aryon n-ary
relations to form new relations.relations to form new relations.
Definition:Definition: TheThe projectionprojection PPii11, i, i22, …, i, …, imm
maps the n-tuplemaps the n-tuple
(a(a11, a, a22, …, a, …, ann) to the m-tuple (a) to the m-tuple (aii11
, a, aii22
, …, a, …, aiimm
), where m), where m ≤≤
n.n.
In other words, a projection PIn other words, a projection Pii11, i, i22, …, i, …, imm
keeps the mkeeps the m
components acomponents aii11
, a, aii22
, …, a, …, aiimm
of an n-tuple and deletes itsof an n-tuple and deletes its
(n – m) other components.(n – m) other components.
Example:Example: What is the result when we apply theWhat is the result when we apply the
projection Pprojection P2,42,4 to the student record (Stevens, 786576,to the student record (Stevens, 786576,
Psych, 2.99) ?Psych, 2.99) ?
Solution:Solution: It is the pair (786576, 2.99).It is the pair (786576, 2.99).

In some cases, applying a projection to an entire tableIn some cases, applying a projection to an entire table
may not only result in fewer columns, but also inmay not only result in fewer columns, but also in
fewer rowsfewer rows..
Why is that?Why is that?
Some records may only have differed in those fieldsSome records may only have differed in those fields
that were deleted, so they becomethat were deleted, so they become identicalidentical, and, and
there is no need to list identical records more thanthere is no need to list identical records more than
once.once.

We can use theWe can use the joinjoin operation to combine two tablesoperation to combine two tables
into one if they share some identical fields.into one if they share some identical fields.
Definition:Definition: Let R be a relation of degree m and S aLet R be a relation of degree m and S a
relation of degree n. Therelation of degree n. The joinjoin JJpp(R, S), where p(R, S), where p ≤≤ mm
and pand p ≤≤ n, is a relation of degree m + n – p thatn, is a relation of degree m + n – p that
consists of all (m + n – p)-tuplesconsists of all (m + n – p)-tuples
(a(a11, a, a22, …, a, …, am-pm-p, c, c11, c, c22, …, c, …, cpp, b, b11, b, b22, …, b, …, bn-pn-p),),
where the m-tuple (awhere the m-tuple (a11, a, a22, …, a, …, am-pm-p, c, c11, c, c22, …, c, …, cpp) belongs) belongs
to R and the n-tuple (cto R and the n-tuple (c11, c, c22, …, c, …, cpp, b, b11, b, b22, …, b, …, bn-pn-p) belongs) belongs
to S.to S.

In other words, to generate Jp(R, S), we have to findIn other words, to generate Jp(R, S), we have to find
all the elements in R whose p last components matchall the elements in R whose p last components match
the p first components of an element in S.the p first components of an element in S.
The new relation contains exactly these matches,The new relation contains exactly these matches,
which are combined to tuples that contain eachwhich are combined to tuples that contain each
matching field only once.matching field only once.

Example:Example: What is JWhat is J11(Y, R), where Y contains the(Y, R), where Y contains the
fieldsfields Student NameStudent Name andand Year of BirthYear of Birth,,
Y = {(1978, Ackermann),Y = {(1978, Ackermann),
(1972, Adams),(1972, Adams),
(1917, Chou),(1917, Chou),
(1984, Goodfriend),(1984, Goodfriend),
(1982, Rao),(1982, Rao),
(1970, Stevens)},(1970, Stevens)},
and R contains the student records as defined before?and R contains the student records as defined before?

Solution:Solution: The resulting relation is:The resulting relation is:
{(1978, Ackermann, 231455, CS, 3.88),{(1978, Ackermann, 231455, CS, 3.88),
(1972, Adams, 888323, Physics, 3.45),(1972, Adams, 888323, Physics, 3.45),
(1917, Chou, 102147, CS, 3.79),(1917, Chou, 102147, CS, 3.79),
(1984, Goodfriend, 453876, Math, 3.45),(1984, Goodfriend, 453876, Math, 3.45),
(1982, Rao, 678543, Math, 3.90),(1982, Rao, 678543, Math, 3.90),
(1970, Stevens, 786576, Psych, 2.99)}(1970, Stevens, 786576, Psych, 2.99)}
Since Y has two fields and R has four, the relationSince Y has two fields and R has four, the relation
JJ11(Y, R) has 2 + 4 – 1 = 5 fields.(Y, R) has 2 + 4 – 1 = 5 fields.

Representing RelationsRepresenting Relations
We already know different ways of representingWe already know different ways of representing
relations. We will now take a closer look at two waysrelations. We will now take a closer look at two ways
of representation:of representation: Zero-one matricesZero-one matrices andand directeddirected
graphsgraphs..
If R is a relation from A = {aIf R is a relation from A = {a11, a, a22, …, a, …, amm} to B =} to B =
{b{b11, b, b22, …, b, …, bnn}, then R can be represented by the zero-}, then R can be represented by the zero-
one matrix Mone matrix MRR = [m= [mijij] with] with
mmijij = 1, if (a= 1, if (aii, b, bjj))∈∈R, andR, and
mmijij = 0, if (a= 0, if (aii, b, bjj))∉∉R.R.
Note that for creating this matrix we first need to listNote that for creating this matrix we first need to list
the elements in A and B in athe elements in A and B in a particular, but arbitraryparticular, but arbitrary
orderorder..

Example:Example: How can we represent the relationHow can we represent the relation
R = {(2, 1), (3, 1), (3, 2)} as a zero-one matrix?R = {(2, 1), (3, 1), (3, 2)} as a zero-one matrix?
Solution:Solution: The matrix MThe matrix MRR is given byis given by










=
11
01
00
RM

What do we know about the matrices representing aWhat do we know about the matrices representing a
relation on a setrelation on a set (a relation from A to A) ?(a relation from A to A) ?
They areThey are squaresquare matrices.matrices.
What do we know about matrices representingWhat do we know about matrices representing
reflexivereflexive relations?relations?
All the elements on theAll the elements on the diagonaldiagonal of such matrices Mof such matrices Mrefref
must bemust be 1s1s..




















=
1
.
.
.
1
1
refM

What do we know about the matrices representingWhat do we know about the matrices representing
symmetric relationssymmetric relations??
These matrices are symmetric, that is, MThese matrices are symmetric, that is, MRR = (M= (MRR))tt
..












=
1101
1001
0010
1101
RM
symmetric matrix,symmetric matrix,
symmetric relation.symmetric relation.












=
0011
0011
0011
0011
RM
non-symmetric matrix,non-symmetric matrix,
non-symmetric relation.non-symmetric relation.

The Boolean operationsThe Boolean operations joinjoin andand meetmeet (you(you
remember?)remember?) can be used to determine the matricescan be used to determine the matrices
representing therepresenting the unionunion and theand the intersectionintersection of twoof two
relations, respectively.relations, respectively.
To obtain theTo obtain the joinjoin of two zero-one matrices, we applyof two zero-one matrices, we apply
the Boolean “or” function to all corresponding elementsthe Boolean “or” function to all corresponding elements
in the matrices.in the matrices.
To obtain theTo obtain the meetmeet of two zero-one matrices, we applyof two zero-one matrices, we apply
the Boolean “and” function to all correspondingthe Boolean “and” function to all corresponding
elements in the matrices.elements in the matrices.

Example:Example: Let the relations R and S be represented byLet the relations R and S be represented by
the matricesthe matrices










=∨=∪
011
111
101
SRSR MMM










=
001
110
101
SM
What are the matrices representing RWhat are the matrices representing R∪∪S and RS and R∩∩S?S?
Solution:Solution: These matrices are given byThese matrices are given by










=∧=∩
000
000
101
SRSR MMM










=
010
001
101
RM

Representing Relations Using MatricesRepresenting Relations Using Matrices
Do you remember theDo you remember the Boolean productBoolean product of two zero-of two zero-
one matrices?one matrices?
Let A = [aLet A = [aijij] be an m] be an m××k zero-one matrix andk zero-one matrix and
B = [bB = [bijij] be a k] be a k××n zero-one matrix.n zero-one matrix.
Then theThen the Boolean productBoolean product of A and B, denoted byof A and B, denoted by
AAοοB, is the mB, is the m××n matrix with (i, j)th entry [cn matrix with (i, j)th entry [cijij], where], where
ccijij = (a= (ai1i1 ∧∧ bb1j1j)) ∨∨ (a(ai2i2 ∧∧ bb2i2i)) ∨∨ …… ∨∨ (a(aikik ∧∧ bbkjkj).).
ccijij = 1 if and only if at least one of the terms= 1 if and only if at least one of the terms
(a(ainin ∧∧ bbnjnj) = 1 for some n; otherwise c) = 1 for some n; otherwise cijij = 0.= 0.

Let us now assume that the zero-one matricesLet us now assume that the zero-one matrices
MMAA = [a= [aijij], M], MBB = [b= [bijij] and M] and MCC = [c= [cijij] represent relations A, B,] represent relations A, B,
and C, respectively.and C, respectively.
Remember:Remember: For MFor MCC = M= MAAοοMMBB we have:we have:
ccijij = 1 if and only if at least one of the terms= 1 if and only if at least one of the terms
(a(ainin ∧∧ bbnjnj) = 1 for some n; otherwise c) = 1 for some n; otherwise cijij = 0.= 0.
In terms of theIn terms of the relationsrelations, this means that C contains a pair, this means that C contains a pair
(x(xii, z, zjj) if and only if there is an element y) if and only if there is an element ynn such that (xsuch that (xii, y, ynn))
is in relation A andis in relation A and
(y(ynn, z, zjj) is in relation B.) is in relation B.
Therefore, C = BTherefore, C = B°°A (A (compositecomposite of A and B).of A and B).

This gives us the following rule:This gives us the following rule:
MMBB°°AA = M= MAAοοMMBB
In other words, the matrix representing theIn other words, the matrix representing the
compositecomposite of relations A and B is theof relations A and B is the BooleanBoolean
productproduct of the matrices representing A and B.of the matrices representing A and B.
Analogously, we can find matrices representing theAnalogously, we can find matrices representing the
powers of relationspowers of relations::
MMRRnn = M= MRR
[n][n]
(n-th(n-th Boolean powerBoolean power).).

Example:Example: Find the matrix representing RFind the matrix representing R22
, where the, where the
matrix representing R is given bymatrix representing R is given by










=
001
110
010
RM
Solution:Solution: The matrix for RThe matrix for R22
is given byis given by










==
010
111
110
]2[
2 RR
MM

Representing Relations Using DigraphsRepresenting Relations Using Digraphs
Definition:Definition: AA directed graphdirected graph, or, or digraphdigraph, consists of, consists of
a set V ofa set V of verticesvertices (or(or nodesnodes) together with a set E of) together with a set E of
ordered pairs of elements of V calledordered pairs of elements of V called edgesedges (or(or arcsarcs).).
The vertex a is called theThe vertex a is called the initial vertexinitial vertex of the edge (a,of the edge (a,
b), and the vertex b is called theb), and the vertex b is called the terminal vertexterminal vertex ofof
this edge.this edge.
We can use arrows to display graphs.We can use arrows to display graphs.

Representing Relations Using DigraphsRepresenting Relations Using Digraphs
Example:Example: Display the digraph with V = {a, b, c, d},Display the digraph with V = {a, b, c, d},
E = {(a, b), (a, d), (b, b), (b, d), (c, a), (c, b), (d, b)}.E = {(a, b), (a, d), (b, b), (b, d), (c, a), (c, b), (d, b)}.
aa
bb
ccdd
An edge of the form (b, b) is called aAn edge of the form (b, b) is called a loop.loop.

Pertemuan 5_Relation Matriks_01 (17)

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