DISCRETE MATHEMATICS Lecture 20

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DISCRETE MATHEMATICSLecture 20

• Dr. Kemal Akkaya
• Department of Computer Science

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8.2 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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Example

• Let R be a relation on NxNxN consisting of
  triples (a,b,c) where a, b, c are integers with a
  lt b lt c.
• Then (1, 2, 3) is an element of R
• (2, 4, 3) is not
• Let R be the relation with 5-tuples (A, N, S, D,
  T). A Airline, N flight number, S starting
  point, D destination, T departure time
• Degree of this Relation is 5
• Domain set of all airlines, all flight numbers,
  all cities, all cities, all times

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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.
• Combinations of domains can also uniquely
  identify n-tuples composite key
• 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

• We may define operations on n-ary relations
• 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

• Let Rooms be the relation with attributes Place,
  Seats, BoardType, Computer A (P, S, B, C)
• R(Fan171, 80, Chalk, Yes), (Lws210, 25, No,
  Yes), (Nek138, 50, Chalk, No), (Agr212, 200, No,
  Yes),
• R1 select ( R, BoardType, Chalk)
• R1(Fan171,80, Chalk, Yes), (Nek138, 50, Chalk,
  No),
• R2 select (A, Computer, Yes)
• select (select(R1), Computer, Yes)
• R2 (Fan171,80, Chalk, Yes),

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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

• Let Rooms be the relation with attributes Place,
  Seats, BoardType, Computer A (P, S, B, C)
• R(Fan171, 80, Chalk, Yes), (Lws210, 25, No,
  Yes), (Nek138, 50, Chalk, No), (Agr212, 200, No,
  Yes),
• R1 project (R, P, S) will produce a relation
  limited to the place and number of seats.
• R1 (Fan171, 80), (Lws210, 25), (Nek138, 50),
  (Agr212, 200),

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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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Join Example

• Prof. Course
• John CS202
• Brian CS340
• Kenny CS220
• Kemal CS215

• Course Class Time
• CS215 FNR1326 11
• CS220 FNR2332 9
• CS340 Quig101 2
• CS202 LWS202 3

R2
R1
What is the result of R1 joining R2 (J(R1, R2))?
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Join Result

• Prof. Course Class Time
• Kemal CS215 FNR1326 11
• Kenny CS220 FNR2332 9
• Brian CS340 Quig101 2
• John CS202 LWS202 3

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8.3 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
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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Example

• 1 0
• 0 1 1
• 0 1 1

M
Reflexive?
Symmetric?
Antisymmetric?
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Digraph Reflexive, Symmetric, Transitive
a
a
b
a
a
b
b
b
reflexive
symmetric
transitive
c
c
d
c
d
d
d
c
R(a,b),(b,c),(c,d),(d,a)
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8.4 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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8.5 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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8.6 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.
• A set A together with a partial order ? on A is
  called a partially ordered set or poset and is
  denoted (A, ?).
• Examples , on real numbers, ?, ? on sets.
• 3 3 Reflexive
• 3 4 but 4 is not 3 Antisymmetric
• 3 4 and 4 5 then 3 5 Transitive

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Total Ordering

• Another example the divides relation on Z.
• Note it is not necessarily the case that either
  a?b or b?a.
• 3 9 but neither 5 7 nor 7 5
• 3 and 9 are comparable but not 5 and 7
• Thats why divides relation is a partial
  ordering
• If we need a total ordering then any two elements
  of a poset (A, ?) should be comparable
• (Z, ) is a totally ordered poset.