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7.1 Binary Relations (2nd pass)and 7.2 n-ary
Relations Longin Jan Latecki

• Slides adapted from Kees van Deemter who adopted
  them from Michael P. Franks Course Based on the
  TextDiscrete Mathematics Its Applications
  (5th Edition)by Kenneth H. Rosen

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7.1 Binary Relations (2nd pass)

• Let A, B be any sets. A binary relation R from A
  to B is a subset of AB.
• E.g., lt can be seen as (n,m) n lt m
• (a,b)?R means that a is related to b (by R)
• Also written as aRb also R(a,b)
• E.g., altb and lt (a,b) both mean (a,b)? lt
• A binary relation R corresponds to a
  characteristic function PRAB?T,F

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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)?AB
  (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 x 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.

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Reflexivity and relatives

• A relation R on A is reflexive iff ?a?A(aRa).
• E.g., the relation (a,b) ab is
  reflexive.
• R is irreflexive iff ?a?A(?aRa)
• Note irreflexive does NOT mean not reflexive,
  which is just ??a?A(aRa).

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Reflexivity and relatives

• Theorem A relation R is irreflexive iff its
  complementary relation R is reflexive.
• Example lt is irreflexive is reflexive.
• Proof trivial

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• Can you think of
• Reflexive relations
• Irreflexive relations
• Involving numbers, propositions or sets?

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Some examples

• Reflexive
• , have same cardinality, ?
• lt, gt, ?, ?, etc.
• Irreflexive
• lt, gt, have different cardinality, ?

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Symmetry relatives

• A binary relation R on A is symmetric iff
  ?a,b((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 asymmetric if
  ?a,b((a,b)?R ? (b,a)?R).
• Examples lt is asymmetric, likes is not.

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Some direct consequences

• Theorems
• R is symmetric iff R R-1,
• R is asymmetric iff R?R-1 is empty.

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Symmetry relatives

• 1. R is symmetric iff R R-1
• Suppose R is symmetric. Then (x,y) ? R ? (y,x)
  ? R ? (x,y) ? R-1
• ? Suppose R R-1 Then (x,y) ? R ? (x,y) ?
  R-1 ? (y,x) ? R

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Symmetry relatives

• 2. R is asymmetric iff R?R-1 is
  empty.(Straightforward application of the
  definitions of asymmetry and R-1)

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Antisymmetry

• Consider the relation x?y
• It is not symmetric. (For instance, 5?6 but not
  6?5)
• It is not asymmetric. (For instance, 5 ?5)
• You might say its nearly symmetric, since the
  only symmetries occur when xy
• This is called antisymmetry

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Antisymmetry

• A binary relation R on A is antisymmetric iff
  ?a,b((a,b)?R ? (b,a)?R? ab).
• Examples ?, ?, ?

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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.
• Example is an ancestor of is transitive.

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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
• Function composition is an example writing fT
  for the function-version of the relation T, we
  have
• S?R(a,c) iff ?b aRb ? bSc iff fR(a)b
  and fS(b)ciff fS?fR(a)c

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Composite Relations

• 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.
• E.g., R1 R R2 R?R R3 R?R?R

e
d
a
c
b
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Composite Relations

• R2 R?R (a, c) (e, c) (b, d) (d, d) (c, c)

e
e
d
d
a
a
c
c
b
b
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7.2 n-ary Relations

• An n-ary relation R on sets A1,,An, is a
  subset R ? A1 An.
• The sets Ai are called the domains of R.
• The degree of R is n.
• A relational database is essentially just a set
  of relations.

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7.2 n-ary Relations

• An n-ary relation R is a straightforward
  generalisation of a binary relation. For example
• 3-ary relations
• a is between b and c
• a gave b to c

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Relational Databases

• A domain Ai is a primary key for the database if
  the relation R is functional in Ai.
• R is functional in the domain Ai if it contains
  at most one n-tuple (, ai ,) for any value ai
  within domain 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 consisting of all n-tuples from R
  that satisfy C. That is,
• sC(R) a?R C(a) T

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Selection Operator Example

• Suppose we have a domain A StudentName
  Standing SocSecNos
• Suppose we define a condition Upperlevel 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 students (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, 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 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 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 can also be sequences of elements.

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Join Example

• Suppose R1 is a teaching assignment table,
  relating Lecturers to Courses.
• Suppose R2 is a room assignment table relating
  Courses to Rooms,Times.
• Then J(R1,R2) is like your class schedule,
  listing (lecturer,course,room,time).
• (For precise definition, see Rosen, p.486)