Relations

1
Relations

• Rosen 5th ed., ch. 7

2
Relations

• Relationships between elements of sets occur very
  often.
• (Employee, Salary)
• (Students, Courses, GPA)
• We use ordered pairs (or n-tuples) of elements
  from the sets to represent relationships.

3
Binary Relations

• A binary relation R from set A to set B, written
  RA?B, is a subset of AB.
• E.g., let lt N?N (n,m) n lt m
• The notation aRb means (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)

4
Example

• A students at UNR, B courses offered
  at UNR
• R relation of students enrolled in
  courses
• (Jason, CS365), (Mary, CS201) are in R
• If Mary does not take CS365, then (Mary, CS365)
  is not in R!
• If CS480 is not being offered, then (Jason,
  CS480), (Mary, CS480) are not in R!

5
Complementary Relations

• Let RA?B be any binary relation.
• Then, RA?B, 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

6
Inverse Relations

• Any binary relation RA?B has an inverse relation
  R-1B?A, defined by R-1 (b,a) (a,b)?R.
• E.g., lt-1 (b,a) altb (b,a) bgta gt.

7
Functions as Relations

• A function fA?B is a relation from A to B
• A relation from A to B is not always a function
  fA?B (e.g., relations could be one-to-many)
• Relations are generalizations of functions!

8
Relations on a Set

• A (binary) relation from a set A to itself is
  called a relation on the set A.

A 1,2,3,4 R(a,b) a divides b
9
Example

• How many relations are there on a set A with n
  elements?

10
Reflexivity

• A relation R on A is reflexive if ?a?A, aRa.
• E.g., the relation (a,b) ab is
  reflexive.
• Is the divide relation on the set of positive
  integers reflexive?

11
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)?R
  ? (b,a)?R.
• lt is antisymmetric, likes is not.

12
Transitivity

• A relation R is transitive iff (for all
  a,b,c) (a,b)?R ? (b,c)?R ? (a,c)?R.
• Examples
• is an ancestor of is transitive.
• Is the divides relation on the set of positive
  integers transitive?

13
Combining Relations

• Two relations can be combined in a similar way to
  combining two sets.

14
Composite Relations

• Let RA?B, and SB?C. Then the composite S?R of
  R and S is defined as
• S?R (a,c) aRb ? bSc
• Function composition f?g is an example.
• The nth power Rn of a relation R on a set A can
  be defined recursively by
• R1 R Rn1 Rn?R for all n0.

15
Example

• R is a relation from 1,2,3 to 1,2,3,4
• R (1,1),(1,4),(2,3),(3,1),(3,4)
• S is a relation from 1,2,3,4 to 0,1,2
• S (1,0),(2,0),(3,1),(3,2),(4,1)
• R?S (1,0),(1,1),(2,1),(2,2),(3,0),(3,1)

16
7.2 n-ary Relations

• An n-ary relation R on sets A1,,An, written
  RA1,,An, is a subset R ? A1
  An.
• The degree of R is n.
• Example R consists of 5-tuples (A,N,S,D,T)
• A airplane flights, N flight
  number,
• S starting point, D destination, T
  departure time

17
Databases

• The time required to manipulate information in a
  database depends in how this information is
  stored.
• Operations add/delete, update, search, combine
  etc.
• Various methods for representing databases have
  been developed.
• We will discuss the relational model.

18
Relational Databases

• A database consists of records, which are
  n-tuples, made up of fields.
• A relational database represents records as an
  n-ary relation R.
• (STUDENT_NAME, ID, MAJOR, GPA)
• Relations are also called tables (e.g.,
  displayed as tables often)

19
Relational Databases

• A domain Ai of an n-ary relation is called
  primary key when no two n-tuples have the same
  value on this domain (e.g., ID)
• A composite key is a subset of domains Ai, Aj,
  such that an n-tuple (,ai,,aj,) is
  determined uniquely for each composite value (ai,
  aj,)?AiAj

20
Selection Operator

• Let R be an n-ary relation and C a condition that
  elements in R may satisfy.
• The selection operator sC maps any (n-ary)
  relation R on A to the n-ary relation of all
  n-tuples from R that satisfy C.

21
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).

22
Projection Operators

• The projection operator maps the
• n-tuple
• to the m-tuple
• In other words, it deletes n-m of the components
  of an n-tuple.

23
Example

• Note that fewer rows may result when a projection
  is applied !

24
More Examples

• 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 (database of cars) to obtain a
  list of model/color combinations available.

ik
25
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, C can also be sequences of elements rather
  than single elements.

26
Example
27
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).

28
SQL Example

• SELECT Departure_Time
• FROM Flights
• WHERE destinationDetroit
• Find projection P5 of the selection of 5-tuples
  that satisfy the constraint destinationDetroit

29
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.

30
Using Zero-One Matrices

• To represent a relation R by a matrix MR
  mij, let mij 1 if (ai,bj)?R, else 0.
• E.g., Joe likes Susan and Mary, Fred likes Mary,
  and Mark likes Sally.
• The 0-1 matrix representationof that
  Likesrelation

31
Zero-One Reflexive, Symmetric

• Terms Reflexive, non-Reflexive, symmetric, 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
Non-reflexivesome 0s on diagonal
Symmetricall identicalacross diagonal
Antisymmetricall 1s are acrossfrom 0s
32
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 RA?B can be represented as a graph
  GR(VGA?B, EGR).

Edge set EG(blue arrows)
GR
MR
Joe
Susan
Fred
Mary
Mark
Sally
Node set VG(black dots)
33
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
Asymmetric, non-antisymmetric
Non-reflexive, non-irreflexive
34
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

35
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 a 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.

36
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
37
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
38
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.

39
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).

40
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 be defined using aRb a
  and b have the same f value, given that f.

41
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,

42
Partitions

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