University of Florida Dept' of Computer

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University of FloridaDept. of Computer
Information Science EngineeringCOT
3100Applications of Discrete StructuresDr.
Michael P. Frank

• Slides for a Course Based on the TextDiscrete
  Mathematics Its Applications (5th Edition)by
  Kenneth H. Rosen

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Module 21Relations

• Rosen 5th ed., ch. 7
• 35 slides

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Binary Relations

• Let A, B be any sets. A binary relation R from A
  to B, written (with signature) RAB, or RA,B,
  is (can be identified with) a subset of the set
  AB.
• E.g., let lt N?N (n,m) n lt m
• The notation a R b or aRb means that (a,b)?R.
• E.g., a lt b means (a,b)? lt
• If aRb we may say a is related to b (by relation
  R),
• or just a relates to b (under relation R).
• A binary relation R corresponds to a predicate
  function PRAB?T,F defined over the 2 sets
  A,B
• e.g., predicate eats (a,b) organism a eats
  food b

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Complementary Relations

• Let RA,B be any binary relation.
• Then, RAB, the complement of R, is the binary
  relation defined by R (a,b) (a,b)?R
  (AB) - R
• Note this is just R if the universe of discourse
  is U AB thus the name complement.
• Note the complement of R is R.

Example lt (a,b) (a,b)?lt (a,b) altb

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Inverse Relations

• Any binary relation RAB has an inverse relation
  R-1BA, defined by R-1 (b,a) (a,b)?R.
• E.g., lt-1 (b,a) altb (b,a) bgta gt.
• E.g., if RPeople?Foods is defined by a R
  b ? a eats b, then b R-1 a ? b is eaten by
  a. (Passive voice.)

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Relations on a Set

• A (binary) relation from a set A to itself is
  called a relation on the set A.
• E.g., the lt relation from earlier was defined
  as a relation on the set N of natural numbers.
• The (binary) identity relation IA on a set A is
  the set (a,a)a?A.

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Reflexivity

• A relation R on A is reflexive if ?a?A, aRa.
• E.g., the relation (a,b) ab is
  reflexive.
• A relation R is irreflexive iff its complementary
  relation R is reflexive.
• Example lt is irreflexive, because is
  reflexive.
• Note irreflexive does NOT mean not reflexive!
• For example the relation likes between people
  is not reflexive, but it is not irreflexive
  either.
• Since not everyone likes themselves, but not
  everyone dislikes themselves either!

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Symmetry Antisymmetry

• A binary relation R on A is symmetric iff R
  R-1, that is, if (a,b)?R ? (b,a)?R.
• E.g., (equality) is symmetric. lt is not.
• is married to is symmetric, likes is not.
• A binary relation R is antisymmetric if ?a?b,
  (a,b)?R ? (b,a)?R.
• Examples lt is antisymmetric, likes is not.
• Exercise prove this definition of antisymmetric
  is equivalent to the statement that R?R-1 ? IA.

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Transitivity

• A relation R is transitive iff (for all
  a,b,c) (a,b)?R ? (b,c)?R ? (a,c)?R.
• A relation is intransitive iff it is not
  transitive.
• Some examples
• is an ancestor of is transitive.
• likes between people is intransitive.
• is located within 1 mile of is ?

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Totality

• A relation RAB is total if for every a?A, there
  is at least one b?B such that (a,b)?R.
• If R is not total, then it is called strictly
  partial.
• A partial relation is a relation that might be
  strictly partial. (Or, it might be total.)
• In other words, all relations are considered
  partial.

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Functionality

• A relation RAB is functional if, for any a?A,
  there is at most 1 b?B such that (a,b)?R.
• R is functional ? ?a?A ?b1?b2?B aRb1 ? aRb2.
  
• Iff R is functional, then it corresponds to a
  partial function RA?B
• where R(a)b ? aRb e.g.
• E.g., The relation aRb a b 0 yields the
  function -(a) b.
• R is antifunctional if its inverse relation R-1
  is functional.
• Note A functional relation (partial function)
  that is also antifunctional is an invertible
  partial function.
• R is a total function RA?B if it is both
  functional and total, that is, for any a?A, there
  is exactly 1 b such that (a,b)?R. I.e., ?a?A
  ?!b aRb.
• If R is functional but not total, then it is a
  strictly partial function.
• Exercise Show that iff R is functional and
  antifunctional, and both it and its inverse are
  total, then it is a bijective function.

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Lambda Notation

• The lambda calculus is a formal mathematical
  language for defining and operating on functions.
• It is the mathematical foundation of a number of
  functional (recursive function-based) programming
  languages, such as LISP and ML.
• It is based on lambda notation, in which ?a
  f(a) is a way to denote the function f without
  ever assigning it a specific symbol.
• E.g., (?x. 3x22x) is a name for the function
  fR?R where f(x)3x22x.
• Lambda notation and the such that operator ?
  can also be used to compose an expression for the
  function that corresponds to any given functional
  relation.
• Let RAB be any functional relation on A,B.
• Then the expression (?a b ? aRb) denotes the
  function fA?B where f(a) b iff aRb.
• E.g., If I write f (?a b ? ab 0),this is
  one way of defining the function f(a)-a.

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Composite Relations

• Let RAB, and SBC. Then the composite S?R of
  R and S is defined as
• S?R (a,c) ?b aRb ? bSc
• Note that function composition f?g is an example.
• Exer. Prove that RAA is transitive iff R?R R
• (actually, its not true give a counter-example
  instead).
• The nth power Rn of a relation R on a set A can
  be defined recursively by R0 IA Rn1
  Rn?R for all n0.
• Negative powers of R can also be defined if
  desired, by R-n (R-1)n.

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7.2 n-ary Relations

• An n-ary relation R on sets A1,,An, written
  (with signature) RA1An or RA1,,An, is
  simply a subset R ? A1 An.
• The sets Ai are called the domains of R.
• The degree of R is n.
• R is functional in the domain Ai if it contains
  at most one n-tuple (, ai ,) for any value ai
  within domain Ai.

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Relational Databases

• A relational database is essentially just an
  n-ary relation R.
• A domain Ai is a primary key for the database if
  the relation R is functional in Ai.
• A composite key for the database is a set of
  domains Ai, Aj, such that R contains at most
  1 n-tuple (,ai,,aj,) for each composite value
  (ai, aj,)?AiAj

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Selection Operators

• Let A be any n-ary domain AA1An, and let
  CA?T,F be any condition (predicate) on
  elements (n-tuples) of A.
• Then, the selection operator sC is the operator
  that maps any (n-ary) relation R on A to the
  n-ary relation of all n-tuples from R that
  satisfy C.
• I.e., ?R?A, sC(R) a?R sC(a) T

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Selection Operator Example

• Suppose we have a domain A StudentName
  Standing SocSecNos
• Suppose we define a certain condition on A,
  UpperLevel(name,standing,ssn) (standing
  junior) ? (standing senior)
• Then, sUpperLevel is the selection operator that
  takes any relation R on A (database of students)
  and produces a relation consisting of just the
  upper-level classes (juniors and seniors).

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Projection Operators

• Let A A1An be any n-ary domain, and let
  ik(i1,,im) be a sequence of indices all
  falling in the range 1 to n,
• That is, where 1 ik n for all 1 k m.
• Then the projection operator on n-tuplesis
  defined by

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Projection Example

• Suppose we have a ternary (3-ary) domain
  CarsModelYearColor. (note n3).
• Consider the index sequence ik 1,3. (m2)
• Then the projection P simply maps each tuple
  (a1,a2,a3) (model,year,color) to its image
• This operator can be usefully applied to a whole
  relation R?Cars (a database of cars) to obtain a
  list of the model/color combinations available.

ik
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Join Operator

• Puts two relations together to form a sort of
  combined relation.
• If the tuple (A,B) appears in R1, and the tuple
  (B,C) appears in R2, then the tuple (A,B,C)
  appears in the join J(R1,R2).
• A, B, and C here can also be sequences of
  elements (across multiple fields), not just
  single elements.

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Join Example

• Suppose R1 is a teaching assignment table,
  relating Professors to Courses.
• Suppose R2 is a room assignment table relating
  Courses to Rooms,Times.
• Then J(R1,R2) is like your class schedule,
  listing (professor,course,room,time).

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7.3 Representing Relations

• Some ways to represent n-ary relations
• With an explicit list or table of its tuples.
• With a function from the domain to T,F.
• Or with an algorithm for computing this function.
• Some special ways to represent binary relations
• With a zero-one matrix.
• With a directed graph.

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Using Zero-One Matrices

• To represent a binary relation RAB by an
  AB 0-1 matrix MR mij, let mij 1 iff
  (ai,bj)?R.
• E.g., Suppose Joe likes Susan and Mary, Fred
  likes Mary, and Mark likes Sally.
• Then the 0-1 matrix representationof the
  relationLikesBoysGirlsrelation is

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Zero-One Reflexive, Symmetric

• Terms Reflexive, non-reflexive,
  irreflexive,symmetric, asymmetric, and
  antisymmetric.
• These relation characteristics are very easy to
  recognize by inspection of the zero-one matrix.

any-thing
any-thing
anything
anything
any-thing
any-thing
Reflexiveall 1s on diagonal
Irreflexiveall 0s on diagonal
Symmetricall identicalacross diagonal
Antisymmetricall 1s are acrossfrom 0s
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Using Directed Graphs

• A directed graph or digraph G(VG,EG) is a set VG
  of vertices (nodes) with a set EG?VGVG of edges
  (arcs,links). Visually represented using dots
  for nodes, and arrows for edges. Notice that a
  relation RAB can be represented as a graph
  GR(VGA?B, EGR).

Edge set EG(blue arrows)
Graph rep. GR
Matrix representation MR
Joe
Susan
Fred
Mary
Mark
Sally
Node set VG(black dots)
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Digraph Reflexive, Symmetric

• It is extremely easy to recognize the
  reflexive/irreflexive/ symmetric/antisymmetric
  properties by graph inspection.

?
?
?
?
?
?
?
?
?
?
?
ReflexiveEvery nodehas a self-loop
IrreflexiveNo nodelinks to itself
SymmetricEvery link isbidirectional
AntisymmetricNo link isbidirectional
These are asymmetric non-antisymmetric
These are non-reflexive non-irreflexive
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7.4 Closures of Relations

• For any property X, the X closure of a set A is
  defined as the smallest superset of A that has
  the given property.
• The reflexive closure of a relation R on A is
  obtained by adding (a,a) to R for each a?A.
  I.e., it is R ? IA
• The symmetric closure of R is obtained by adding
  (b,a) to R for each (a,b) in R. I.e., it is R ?
  R-1
• The transitive closure or connectivity relation
  of R is obtained by repeatedly adding (a,c) to R
  for each (a,b),(b,c) in R.
• I.e., it is

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Paths in Digraphs/Binary Relations

• A path of length n from node a to b in the
  directed graph G (or the binary relation R) is a
  sequence (a,x1), (x1,x2), , (xn-1,b) of n
  ordered pairs in EG (or R).
• An empty sequence of edges is considered a path
  of length 0 from a to a.
• If any path from a to b exists, then we say that
  a is connected to b. (You can get there from
  here.)
• A path of length n1 from a to itself is called a
  circuit or a cycle.
• Note that there exists a path of length n from a
  to b in R if and only if (a,b)?Rn.

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Simple Transitive Closure Alg.

• A procedure to compute R with 0-1 matrices.
• procedure transClosure(MRrank-n 0-1 mat.)
• A B MR
• for i 2 to n begin A A?MR B B ? A
  joinendreturn B Alg. takes T(n4) time

note A represents Ri, B represents
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A Faster Transitive Closure Alg.

• procedure transClosure(MRrank-n 0-1 mat.)
• A MR
• for i 1 to ?log2 n? begin A A?(AIn) A
  represents endreturn A Alg. takes
  only T(n3 log n) time!

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Roy-Warshall Algorithm

• Uses only T(n3) operations!
• Procedure Warshall(MR rank-n 0-1 matrix)
• W MR
• for k 1 to n for i 1 to n for j 1 to
  n wij wij ? (wik ? wkj)return W this
  represents R

wij 1 means there is a path from i to j going
only through nodes k
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7.5 Equivalence Relations

• An equivalence relation (e.r.) on a set A is
  simply any binary relation on A that is
  reflexive, symmetric, and transitive.
• E.g., itself is an equivalence relation.
• For any function fA?B, the relation have the
  same f value, or f (a1,a2) f(a1)f(a2)
  is an equivalence relation,
• e.g., let mmother of then m have the same
  mother is an e.r.

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Equivalence Relation Examples

• Strings a and b are the same length.
• Integers a and b have the same absolute value.
• Real numbers a and b have the same fractional
  part. (i.e., a - b ? Z)
• Integers a and b have the same residue modulo
  m. (for a given mgt1)

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Equivalence Classes

• Let R be any equiv. rel. on a set A.
• The equivalence class of a, aR b aRb
  (optional subscript R)
• It is the set of all elements of A that are
  equivalent to a according to the eq.rel. R.
• Each such b (including a itself) is called a
  representative of aR.
• Since f(a)aR is a function of a, any
  equivalence relation R can be defined using aRb
  a and b have the same f value, given f.

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Equivalence Class Examples

• Strings a and b are the same length.
• a the set of all strings of the same length
  as a.
• Integers a and b have the same absolute value.
• a the set a, -a
• Real numbers a and b have the same fractional
  part (i.e., a - b ? Z).
• a the set , a-2, a-1, a, a1, a2,
• Integers a and b have the same residue modulo
  m. (for a given mgt1)
• a the set , a-2m, a-m, a, am, a2m,

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Partitions

• A partition of a set A is the set of all the
  equivalence classes A1, A2, for some
  equivalence relation on A.
• The Ais are all disjoint and their union A.
• They partition the set into pieces. Within
  each piece, all members of that set are
  equivalent to each other.

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7.6 Partial Orderings

• A relation R on A is called a partial ordering or
  partial order iff it is reflexive, antisymmetric,
  and transitive.
• We often use a symbol looking something like ?
  (or analogous shapes) for such relations.
• Examples , on real numbers, ?, ? on sets.
• Another example the divides relation on Z.
• Note it is not necessarily the case that either
  a?b or b?a.
• A set A together with a partial order ? on A is
  called a partially ordered set or poset and is
  denoted (A, ?).

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Posets as Noncyclical Digraphs

• There is a one-to-one correspondence between
  posets and the reflexivetransitive closures of
  noncyclical digraphs.
• Noncyclical Containing no directed cycles.
• Example consider the poset (0,,10,)
• Its minimaldigraph

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