Sadat sumon

Fall 2002 CMSC 203 - Discrete Structures 1
You Never Escape Your…You Never Escape Your…
RelationsRelations
ByBy
Sadat SumonSadat Sumon

RelationsRelations
If we want to describe a relationship betweenIf we want to describe a relationship between
elements of two sets A and B, we can useelements of two sets A and B, we can use orderedordered
pairspairs with their first element taken from A andwith their first element taken from A and
their second element taken from B.their second element taken from B.
Since this is a relation betweenSince this is a relation between two setstwo sets, it is, it is
called acalled a binary relationbinary relation..
Definition:Definition: Let A and B be sets. A binary relationLet A and B be sets. A binary relation
from A to B is a subset of Afrom A to B is a subset of A××B.B.
In other words, for a binary relation R we haveIn other words, for a binary relation R we have
RR ⊆⊆ AA××B. We use the notation aRb to denote thatB. We use the notation aRb to denote that
(a, b)(a, b)∈∈R and aR and aRb to denote that (a, b)b to denote that (a, b)∉∉R.R.

RelationsRelations
When (a, b) belongs to R, a is said to beWhen (a, b) belongs to R, a is said to be relatedrelated toto
b by R.b by R.
Example:Example: Let P be a set of people, C be a set ofLet P be a set of people, C be a set of
cars, and D be the relation describing which personcars, and D be the relation describing which person
drives which car(s).drives which car(s).
P = {Carl, Suzanne, Peter, Carla},P = {Carl, Suzanne, Peter, Carla},
C = {Mercedes, BMW, tricycle}C = {Mercedes, BMW, tricycle}
D = {(Carl, Mercedes), (Suzanne, Mercedes),D = {(Carl, Mercedes), (Suzanne, Mercedes),
(Suzanne, BMW), (Peter, tricycle)}(Suzanne, BMW), (Peter, tricycle)}
This means that Carl drives a Mercedes, SuzanneThis means that Carl drives a Mercedes, Suzanne
drives a Mercedes and a BMW, Peter drives adrives a Mercedes and a BMW, Peter drives a
tricycle, and Carla does not drive any of thesetricycle, and Carla does not drive any of these
vehicles.vehicles.

Functions as RelationsFunctions as Relations
You might remember that aYou might remember that a functionfunction f from a set Af from a set A
to a set B assigns a unique element of B to eachto a set B assigns a unique element of B to each
element of A.element of A.
TheThe graphgraph of f is the set of ordered pairs (a, b)of f is the set of ordered pairs (a, b)
such that b = f(a).such that b = f(a).
Since the graph of f is a subset of ASince the graph of f is a subset of A××B, it is aB, it is a
relationrelation from A to B.from A to B.
Moreover, for each elementMoreover, for each element aa of A, there isof A, there is
exactly one ordered pair in the graph that hasexactly one ordered pair in the graph that has aa asas
its first element.its first element.

Functions as RelationsFunctions as Relations
Conversely, if R is a relation from A to B such thatConversely, if R is a relation from A to B such that
every element in A is the first element of exactlyevery element in A is the first element of exactly
one ordered pair of R, then a function can beone ordered pair of R, then a function can be
defined with R as its graph.defined with R as its graph.
This is done by assigning to an element aThis is done by assigning to an element a∈∈A theA the
unique element bunique element b∈∈B such that (a, b)B such that (a, b)∈∈R.R.

Relations on a SetRelations on a Set
Definition:Definition: A relation on the set A is a relationA relation on the set A is a relation
from A to A.from A to A.
In other words, a relation on the set A is a subsetIn other words, a relation on the set A is a subset
of Aof A××A.A.
Example:Example: Let A = {1, 2, 3, 4}. Which ordered pairsLet A = {1, 2, 3, 4}. Which ordered pairs
are in the relation R = {(a, b) | a < b} ?are in the relation R = {(a, b) | a < b} ?

Solution:Solution: R = {R = {(1, 2),(1, 2), (1, 3),(1, 3), (1, 4),(1, 4), (2, 3),(2, 3),(2, 4),(2, 4),(3, 4)}(3, 4)}
44
33
22
11
44332211RR11 11
22
33
44
22
33
44
XX XX XX
XX XX
XX

How many different relations can we define onHow many different relations can we define on
a set A with n elements?a set A with n elements?
A relation on a set A is a subset of AA relation on a set A is a subset of A××A.A.
How many elements are in AHow many elements are in A××A ?A ?
There are nThere are n22
elements in Aelements in A××A, so how manyA, so how many
subsets (= relations on A) does Asubsets (= relations on A) does A××A have?A have?
The number of subsets that we can form out of aThe number of subsets that we can form out of a
set with m elements is 2set with m elements is 2mm
. Therefore, 2. Therefore, 2nn22
subsetssubsets
can be formed out of Acan be formed out of A××A.A.
Answer:Answer: We can define 2We can define 2nn22
different relationsdifferent relations
on A.on A.

Properties of RelationsProperties of Relations
We will now look at some useful ways to classifyWe will now look at some useful ways to classify
relations.relations.
Definition:Definition: A relation R on a set A is calledA relation R on a set A is called
reflexivereflexive if (a, a)if (a, a)∈∈R for every element aR for every element a∈∈A.A.
Are the following relations on {1, 2, 3, 4} reflexive?Are the following relations on {1, 2, 3, 4} reflexive?
R = {(1, 1), (1, 2), (2, 3), (3, 3), (4, 4)}R = {(1, 1), (1, 2), (2, 3), (3, 3), (4, 4)} No.No.
R = {(1, 1), (2, 2), (2, 3), (3, 3), (4, 4)}R = {(1, 1), (2, 2), (2, 3), (3, 3), (4, 4)} Yes.Yes.
R = {(1, 1), (2, 2), (3, 3)}R = {(1, 1), (2, 2), (3, 3)} No.No.
Definition:Definition: A relation on a set A is calledA relation on a set A is called
irreflexiveirreflexive if (a, a)if (a, a)∉∉R for every element aR for every element a∈∈A.A.

Definitions:Definitions:
A relation R on a set A is calledA relation R on a set A is called symmetricsymmetric if (b,if (b,
a)a)∈∈R whenever (a, b)R whenever (a, b)∈∈R for all a, bR for all a, b∈∈A.A.
A relation R on a set A is calledA relation R on a set A is called antisymmetricantisymmetric ifif
a = b whenever (a, b)a = b whenever (a, b)∈∈R and (b, a)R and (b, a)∈∈R.R.
A relation R on a set A is calledA relation R on a set A is called asymmetricasymmetric ifif
(a, b)(a, b)∈∈R implies that (b, a)R implies that (b, a)∉∉R for all a, bR for all a, b∈∈A.A.

Are the following relations on {1, 2, 3, 4}Are the following relations on {1, 2, 3, 4}
symmetric, antisymmetric, or asymmetric?symmetric, antisymmetric, or asymmetric?
R = {(1, 1), (1, 2), (2, 1), (3, 3), (4, 4)}R = {(1, 1), (1, 2), (2, 1), (3, 3), (4, 4)} symmetricsymmetric
R = {(1, 1)}R = {(1, 1)} sym. andsym. and
antisym.antisym.
R = {(1, 3), (3, 2), (2, 1)}R = {(1, 3), (3, 2), (2, 1)} antisym.antisym.
and asym.and asym.
R = {(4, 4), (3, 3), (1, 4)}R = {(4, 4), (3, 3), (1, 4)} antisym.antisym.

transitivetransitive if whenever (a, b)if whenever (a, b)∈∈R and (b, c)R and (b, c)∈∈R, thenR, then
(a, c)(a, c)∈∈R for a, b, cR for a, b, c∈∈A.A.
Are the following relations on {1, 2, 3, 4}Are the following relations on {1, 2, 3, 4}
transitive?transitive?
R = {(1, 1), (1, 2), (2, 2), (2, 1), (3, 3)}R = {(1, 1), (1, 2), (2, 2), (2, 1), (3, 3)} Yes.Yes.
R = {(1, 3), (3, 2), (2, 1)}R = {(1, 3), (3, 2), (2, 1)} No.No.
R = {(2, 4), (4, 3), (2, 3), (4, 1)}R = {(2, 4), (4, 3), (2, 3), (4, 1)} No.No.

Counting RelationsCounting Relations
Example:Example: How many different reflexive relationsHow many different reflexive relations
can be defined on a set A containing n elements?can be defined on a set A containing n elements?
Solution:Solution: Relations on R are subsets of ARelations on R are subsets of A××A, whichA, which
contains ncontains n22
elements.elements.
Therefore, different relations on A can beTherefore, different relations on A can be
generated by choosing different subsets out ofgenerated by choosing different subsets out of
these nthese n22
elements, so there are 2elements, so there are 2nn22
AA reflexivereflexive relation, however,relation, however, mustmust contain the ncontain the n
elements (a, a) for every aelements (a, a) for every a∈∈A.A.
Consequently, we can only choose among nConsequently, we can only choose among n22
– n =– n =
n(n – 1) elements to generate reflexive relations, son(n – 1) elements to generate reflexive relations, so
there are 2there are 2n(n – 1)n(n – 1)
of them.of them.

Combining RelationsCombining Relations
Relations are sets, and therefore, we can apply theRelations are sets, and therefore, we can apply the
usualusual set operationsset operations to them.to them.
If we have two relations RIf we have two relations R11 and Rand R22, and both of, and both of
them are from a set A to a set B, then we canthem are from a set A to a set B, then we can
combine them to Rcombine them to R11 ∪∪ RR22, R, R11 ∩∩ RR22, or R, or R11 – R– R22..
In each case, the result will beIn each case, the result will be another relationanother relation
from A to Bfrom A to B..

…… and there is another important way to combineand there is another important way to combine
Definition:Definition: Let R be a relation from a set A to aLet R be a relation from a set A to a
set B and S a relation from B to a set C. Theset B and S a relation from B to a set C. The
compositecomposite of R and S is the relation consisting ofof R and S is the relation consisting of
ordered pairs (a, c), where aordered pairs (a, c), where a∈∈A, cA, c∈∈C, and for whichC, and for which
there exists an element bthere exists an element b∈∈B such that (a, b)B such that (a, b)∈∈R andR and
(b, c)(b, c)∈∈S. We denote the composite of R and S byS. We denote the composite of R and S by
SS°° RR..
In other words, if relation R contains a pair (a, b)In other words, if relation R contains a pair (a, b)
and relation S contains a pair (b, c), then Sand relation S contains a pair (b, c), then S°° RR
contains a pair (a, c).contains a pair (a, c).

Example:Example: Let D and S be relations on A = {1, 2, 3, 4}.Let D and S be relations on A = {1, 2, 3, 4}.
D = {(a, b) | b = 5 - a} “b equals (5 – a)”D = {(a, b) | b = 5 - a} “b equals (5 – a)”
S = {(a, b) | a < b} “a is smaller than b”S = {(a, b) | a < b} “a is smaller than b”
D = {(1, 4), (2, 3), (3, 2), (4, 1)}D = {(1, 4), (2, 3), (3, 2), (4, 1)}
S = {(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)}S = {(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)}
SS°° D = {D = { (2, 4),(2, 4), (3, 3),(3, 3), (3, 4),(3, 4), (4, 2),(4, 2), (4, 3),(4, 3),
D maps an element a to the element (5 – a), andD maps an element a to the element (5 – a), and
afterwards S maps (5 – a) to all elements larger thanafterwards S maps (5 – a) to all elements larger than
(5 – a), resulting in(5 – a), resulting in SS°° D = {(a,b) | b > 5 – a}D = {(a,b) | b > 5 – a} oror SS°° DD
= {(a,b) | a + b > 5}.= {(a,b) | a + b > 5}.
(4, 4)}(4, 4)}

We already know thatWe already know that functionsfunctions are justare just specialspecial
casescases ofof relationsrelations (namely those that map each(namely those that map each
element in the domain onto exactly one element inelement in the domain onto exactly one element in
the codomain).the codomain).
If we formally convert two functions into relations,If we formally convert two functions into relations,
that is, write them down as sets of ordered pairs,that is, write them down as sets of ordered pairs,
the composite of these relations will be exactly thethe composite of these relations will be exactly the
same as the composite of the functions (as definedsame as the composite of the functions (as defined
earlier).earlier).

Definition:Definition: Let R be a relation on the set A. TheLet R be a relation on the set A. The
powers Rpowers Rnn
, n = 1, 2, 3, …, are defined inductively by, n = 1, 2, 3, …, are defined inductively by
RR11
= R= R
RRn+1n+1
= R= Rnn
°° RR
In other words:In other words:
RRnn
= R= R°° RR°° …… °° R (n times the letter R)R (n times the letter R)

Theorem:Theorem: The relation R on a set A is transitive ifThe relation R on a set A is transitive if
and only if Rand only if Rnn
⊆⊆ R for all positive integers n.R for all positive integers n.
Remember the definition of transitivity:Remember the definition of transitivity:
transitive if whenever (a, b)transitive if whenever (a, b)∈∈R and (b, c)R and (b, c)∈∈R, thenR, then
(a, c)(a, c)∈∈R for a, b, cR for a, b, c∈∈A.A.
The composite of R with itself contains exactlyThe composite of R with itself contains exactly
these pairs (a, c).these pairs (a, c).
Therefore, for a transitive relation R, RTherefore, for a transitive relation R, R°° R does notR does not
contain any pairs that are not in R, so Rcontain any pairs that are not in R, so R°° RR ⊆⊆ R.R.
Since RSince R°° R does not introduce any pairs that are notR does not introduce any pairs that are not
already in R, it must also be true that (Ralready in R, it must also be true that (R°° R)R)°° RR ⊆⊆ R,R,
and so on, so that Rand so on, so that Rnn
⊆⊆ R.R.

n-ary Relationsn-ary Relations
In order to study an interesting application ofIn order to study an interesting application of
relations, namelyrelations, namely databasesdatabases, we first need to, we first need to
generalize the concept of binary relations togeneralize the concept of binary relations to n-aryn-ary
relationsrelations..
Definition:Definition: Let ALet A11, A, A22, …, A, …, Ann be sets. Anbe sets. An n-aryn-ary
relationrelation 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∈∈NN}}
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, namelyrepresentation that is based on relations, namely
thethe relational data model.relational data model.
A database consists of n-tuples calledA database consists of n-tuples called recordsrecords,,
which are made up ofwhich are 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 primaryprimary
keykey if the n-tuples are uniquely determined byif the n-tuples are uniquely determined by
their values from this domain.their values from this domain.
This means that no two records have the sameThis means that no two records have the same
value from the same primary key.value from the same primary key.
In our example, which of the fieldsIn our example, which of the fields Student NameStudent Name,,
ID NumberID Number,, 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 wouldwould
be a primary key.be a primary key.

In a database, a primary key should remain oneIn a database, a primary key should remain one
even if new records are added.even 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-of the database, containing all the n-
tuples that can ever be included in our database.tuples 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 andetermine an
n-tuple in a relation, then-tuple in a relation, the Cartesian productCartesian product ofof
these domains is called athese domains 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,to the student record (Stevens,
786576, Psych, 2.99) ?786576, 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 entireIn some cases, applying a projection to an entire
table may not only result in fewer columns, but alsotable may not only result in fewer columns, but also
inin fewer rowsfewer rows..
Why is that?Why is that?
Some records may only have differed in thoseSome records may only have differed in those
fields that were deleted, so they becomefields that were deleted, so they become identicalidentical,,
and there is no need to list identical records moreand there is no need to list identical records more
than once.than once.

We can use theWe can use the joinjoin operation to combine twooperation to combine two
tables into one if they share some identical fields.tables 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 to S.belongs to S.

In other words, to generate Jp(R, S), we have toIn other words, to generate Jp(R, S), we have to
find all the elements in R whose p last componentsfind all the elements in R whose p last components
match the p first components of an element in S.match 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 definedand R contains the student records as defined
before ?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 tworelations. We will now take a closer look at two
ways of representation:ways of representation: Zero-one matricesZero-one matrices andand
directed graphsdirected graphs..
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}, then R can be represented by the
zero-one matrix Mzero-one 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 toNote that for creating this matrix we first need to
list the elements in A and B in alist the elements in A and B in a particular, butparticular, but
arbitrary orderarbitrary order..

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 representingWhat do we know about the matrices representing
aa 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 matricesof such matrices
MMrefref 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, weof two zero-one matrices, we
apply the Boolean “or” function to all correspondingapply the Boolean “or” function to all corresponding
elements in the matrices.elements in the matrices.
To obtain theTo obtain the meetmeet of two zero-one matrices, weof two zero-one matrices, we
apply the Boolean “and” function to all correspondingapply the Boolean “and” function to all corresponding
elements in the matrices.elements in the matrices.

Example:Example: Let the relations R and S be representedLet the relations R and S be represented
by the matricesby the 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
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

Example:Example: Let the relations R and S be representedLet the relations R and S be represented
by the matricesby the 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

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,] represent relations A,
B, and C, respectively.B, 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, this means that C contains a
pair (xpair (xii, z, zjj) if and only if there is an element y) if and only if there is an element ynn suchsuch
that (xthat (xii, y, ynn) is in relation A and) is 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, where
the matrix representing R is given bythe matrix 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, consists
of a set V ofof a set V of verticesvertices (or(or nodesnodes) together with a) together with a
set E of ordered pairs of elements of V calledset E of ordered 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 theof the
edge (a, b), and the vertex b is called theedge (a, b), and the vertex b is called the terminalterminal
vertexvertex of this edge.of this edge.
We can use arrows to display graphs.We can use arrows to display graphs.

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.

Obviously, we can represent any relation R on a setObviously, we can represent any relation R on a set
A by the digraph with A as its vertices and all pairsA by the digraph with A as its vertices and all pairs
(a, b)(a, b)∈∈R as its edges.R as its edges.
Vice versa, any digraph with vertices V and edges EVice versa, any digraph with vertices V and edges E
can be represented by a relation on V containing allcan be represented by a relation on V containing all
the pairs in E.the pairs in E.
ThisThis one-to-one correspondenceone-to-one correspondence betweenbetween
relations and digraphs means that any statementrelations and digraphs means that any statement
about relations also applies to digraphs, and viceabout relations also applies to digraphs, and vice
versa.versa.

Equivalence RelationsEquivalence Relations
Equivalence relationsEquivalence relations are used to relate objectsare used to relate objects
that are similar in some way.that are similar in some way.
Definition:Definition: A relation on a set A is called anA relation on a set A is called an
equivalence relation if it is reflexive, symmetric,equivalence relation if it is reflexive, symmetric,
and transitive.and transitive.
Two elements that are related by an equivalenceTwo elements that are related by an equivalence
relation R are calledrelation R are called equivalentequivalent..

Since R isSince R is symmetricsymmetric, a is equivalent to b whenever, a is equivalent to b whenever
b is equivalent to a.b is equivalent to a.
Since R isSince R is reflexivereflexive, every element is equivalent to, every element is equivalent to
itself.itself.
Since R isSince R is transitivetransitive, if a and b are equivalent and b, if a and b are equivalent and b
and c are equivalent, then a and c are equivalent.and c are equivalent, then a and c are equivalent.
Obviously, these three properties are necessaryObviously, these three properties are necessary
for a reasonable definition of equivalence.for a reasonable definition of equivalence.

Example:Example: Suppose that R is the relation on the setSuppose that R is the relation on the set
of strings that consist of English letters such thatof strings that consist of English letters such that
aRb if and only if l(a) = l(b), where l(x) is the lengthaRb if and only if l(a) = l(b), where l(x) is the length
of the string x. Is R an equivalence relation?of the string x. Is R an equivalence relation?
Solution:Solution:
• R is reflexive, because l(a) = l(a) and thereforeR is reflexive, because l(a) = l(a) and therefore
aRa for any string a.aRa for any string a.
• R is symmetric, because if l(a) = l(b) then l(b) =R is symmetric, because if l(a) = l(b) then l(b) =
l(a), so if aRb then bRa.l(a), so if aRb then bRa.
• R is transitive, because if l(a) = l(b) and l(b) = l(c),R is transitive, because if l(a) = l(b) and l(b) = l(c),
then l(a) = l(c), so aRb and bRc implies aRc.then l(a) = l(c), so aRb and bRc implies aRc.
R is an equivalence relation.R is an equivalence relation.

Equivalence ClassesEquivalence Classes
Definition:Definition: Let R be an equivalence relation on aLet R be an equivalence relation on a
set A. The set of all elements that are related toset A. The set of all elements that are related to
an element a of A is called thean element a of A is called the equivalence classequivalence class
of a.of a.
The equivalence class of a with respect to R isThe equivalence class of a with respect to R is
denoted bydenoted by [a][a]RR..
When only one relation is under consideration, weWhen only one relation is under consideration, we
will delete the subscript R and writewill delete the subscript R and write [a][a] for thisfor this
equivalence class.equivalence class.
If bIf b∈∈[a][a]RR, b is called a, b is called a representativerepresentative of thisof this
equivalence class.equivalence class.

Example:Example: In the previous example (strings ofIn the previous example (strings of
identical length), what is the equivalence class ofidentical length), what is the equivalence class of
the word mouse, denoted by [mouse] ?the word mouse, denoted by [mouse] ?
Solution:Solution: [mouse] is the set of all English words[mouse] is the set of all English words
containing five letters.containing five letters.
For example, ‘horse’ would be a representative ofFor example, ‘horse’ would be a representative of
this equivalence class.this equivalence class.

Theorem:Theorem: Let R be an equivalence relation on a setLet R be an equivalence relation on a set
A. The following statements are equivalent:A. The following statements are equivalent:
• aRbaRb
• [a] = [b][a] = [b]
• [a][a] ∩∩ [b][b] ≠≠ ∅∅
Definition:Definition: AA partitionpartition of a set S is a collection ofof a set S is a collection of
disjoint nonempty subsets of S that have S as theirdisjoint nonempty subsets of S that have S as their
union. In other words, the collection of subsets Aunion. In other words, the collection of subsets Aii,,
ii∈∈I, forms a partition of S if and only ifI, forms a partition of S if and only if
(i) A(i) Aii ≠≠ ∅∅ for ifor i∈∈II
• AAii ∩∩ AAjj == ∅∅, if i, if i ≠≠ jj
• ∪∪ii∈∈II AAii = S= S

Examples:Examples: Let S be the set {u, m, b, r, o, c, k, s}.Let S be the set {u, m, b, r, o, c, k, s}.
Do the following collections of sets partition S ?Do the following collections of sets partition S ?
{{m, o, c, k}, {r, u, b, s}}{{m, o, c, k}, {r, u, b, s}} yes.yes.
{{c, o, m, b}, {u, s}, {r}}{{c, o, m, b}, {u, s}, {r}} no (k is missing).no (k is missing).
{{b, r, o, c, k}, {m, u, s, t}}{{b, r, o, c, k}, {m, u, s, t}} no (t is not in S).no (t is not in S).
{{u, m, b, r, o, c, k, s}}{{u, m, b, r, o, c, k, s}} yes.yes.
{{b, o, o, k}, {r, u, m}, {c, s}}{{b, o, o, k}, {r, u, m}, {c, s}} yes ({b,o,o,k} = {b,o,k}).yes ({b,o,o,k} = {b,o,k}).
{{u, m, b}, {r, o, c, k, s},{{u, m, b}, {r, o, c, k, s}, ∅∅}} no (no (∅∅ not allowed).not allowed).

Theorem:Theorem: Let R be an equivalence relation on aLet R be an equivalence relation on a
set S. Then theset S. Then the equivalence classesequivalence classes of R form aof R form a
partitionpartition of S. Conversely, given a partitionof S. Conversely, given a partition
{A{Aii | i| i∈∈I} of the set S, there is an equivalenceI} of the set S, there is an equivalence
relation R that has the sets Arelation R that has the sets Aii, i, i∈∈I, as itsI, as its
equivalence classes.equivalence classes.

Example:Example: Let us assume that Frank, Suzanne andLet us assume that Frank, Suzanne and
George live in Boston, Stephanie and Max live inGeorge live in Boston, Stephanie and Max live in
Lübeck, and Jennifer lives in Sydney.Lübeck, and Jennifer lives in Sydney.
Let R be theLet R be the equivalence relationequivalence relation {(a, b) | a and b{(a, b) | a and b
live in the same city} on the set P = {Frank,live in the same city} on the set P = {Frank,
Suzanne, George, Stephanie, Max, Jennifer}.Suzanne, George, Stephanie, Max, Jennifer}.
Then R = {(Frank, Frank), (Frank, Suzanne),Then R = {(Frank, Frank), (Frank, Suzanne),
(Frank, George), (Suzanne, Frank), (Suzanne,(Frank, George), (Suzanne, Frank), (Suzanne,
Suzanne), (Suzanne, George), (George, Frank),Suzanne), (Suzanne, George), (George, Frank),
(George, Suzanne), (George, George), (Stephanie,(George, Suzanne), (George, George), (Stephanie,
Stephanie), (Stephanie, Max), (Max, Stephanie),Stephanie), (Stephanie, Max), (Max, Stephanie),
(Max, Max), (Jennifer, Jennifer)}.(Max, Max), (Jennifer, Jennifer)}.

Then theThen the equivalence classesequivalence classes of R are:of R are:
{{Frank, Suzanne, George}, {Stephanie, Max},{{Frank, Suzanne, George}, {Stephanie, Max},
{Jennifer}}.{Jennifer}}.
This is aThis is a partitionpartition of P.of P.
The equivalence classes of any equivalence relationThe equivalence classes of any equivalence relation
R defined on a set S constitute a partition of S,R defined on a set S constitute a partition of S,
because every element in S is assigned tobecause every element in S is assigned to exactlyexactly
oneone of the equivalence classes.of the equivalence classes.

Another example:Another example: Let R be the relationLet R be the relation
{(a, b) | a{(a, b) | a ≡≡ b (mod 3)} on the set of integers.b (mod 3)} on the set of integers.
Is R an equivalence relation?Is R an equivalence relation?
Yes, R is reflexive, symmetric, and transitive.Yes, R is reflexive, symmetric, and transitive.
What are the equivalence classes of R ?What are the equivalence classes of R ?
{{…, -6, -3, 0, 3, 6, …},{{…, -6, -3, 0, 3, 6, …},
{…, -5, -2, 1, 4, 7, …},{…, -5, -2, 1, 4, 7, …},
{…, -4, -1, 2, 5, 8, …}}{…, -4, -1, 2, 5, 8, …}}

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