Bogazici University Department of Computer Engineering CmpE 220 Discrete Mathematics 21' Relations H - PowerPoint PPT Presentation

1 / 41
About This Presentation
Title:

Bogazici University Department of Computer Engineering CmpE 220 Discrete Mathematics 21' Relations H

Description:

A relation R is irreflexive. iff its complementary relation R is reflexive. Ex. ... 'likes' between people is not reflexive, but it is not irreflexive either. ... – PowerPoint PPT presentation

Number of Views:94
Avg rating:3.0/5.0
Slides: 42
Provided by: michae1484
Category:

less

Transcript and Presenter's Notes

Title: Bogazici University Department of Computer Engineering CmpE 220 Discrete Mathematics 21' Relations H


1
Bogazici UniversityDepartment of Computer
EngineeringCmpE 220 Discrete Mathematics21.
Relations Haluk Bingöl
2
Module 21Relations
  • Rosen 5th ed., ch. 7
  • 35 slides (in progress), 2 lectures

3
Binary Relations
  • Def. 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).
  • Ex. 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

4
Complementary Relations
  • Def. 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

5
Inverse Relations
  • Def. Any binary relation RAB has an inverse
    relation R-1BA, defined by R-1 (b,a)
    (a,b)?R.
  • Ex.lt-1 (b,a) altb (b,a) bgta gt.
  • Ex. If RPeople?Foods is defined by a R
    b ? a eats b, then b R-1 a ? b is eaten by
    a. (Passive voice.)

6
Relations on a Set
  • Def. A (binary) relation from a set A to itself
    is called a relation on the set A.
  • Ex.
  • lt ? NN, lt (n,m) n lt m, n,m?N
  • IA ? AA, IA (a,a)a?A.

7
Reflexivity
  • Def. A relation R on A is reflexive if ?a?A,
    aRa.
  • Ex. (a,b) ab is reflexive.
  • Def. A relation R is irreflexive iff its
    complementary relation R is reflexive.
  • Ex. lt is irreflexive, because is reflexive.
  • Remark. irreflexive does NOT mean not
    reflexive!
  • Ex.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!

8
Symmetry Antisymmetry
  • Def. A binary relation R on A is symmetric iff R
    R-1, that is, if (a,b)?R ? (b,a)?R.
  • Ex. (equality) is symmetric. lt is not.
  • is married to is symmetric, likes is not.
  • Def. A binary relation R is antisymmetric if
    ?a?b, (a,b)?R ? (b,a)?R.
  • Ex.
  • lt is antisymmetric, likes is not.
  • Prove this definition of antisymmetric is
    equivalent to the statement that R?R-1 ? IA.

9
Transitivity
  • Def. A relation R is transitive iff (for all
    a,b,c) (a,b)?R ? (b,c)?R ? (a,c)?R.
  • Def. A relation is intransitive iff it is not
    transitive.
  • Ex.
  • is an ancestor of is transitive.
  • likes between people is intransitive.
  • is located within 1 mile of is ?

10
Totality
  • Def. A relation RAB is total if ?a?A, b?B ?
    (a,b)?R.
  • Def. If R is not total, then it is called
    strictly partial.
  • Def. A partial relation is a relation that might
    be strictly partial. (Or, it might be total.)
  • In other words, all relations are considered
    partial.

11
Functionality
  • Def. 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.
  • Ex. The relation aRb a b 0 yields the
    function -(a) b.
  • Def. 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.
  • Def. 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.
  • Ex. Show that iff R is functional and
    antifunctional, and both it and its inverse are
    total, then it is a bijective function.

12
Lambda Notation
  • Def. 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.

13
Composite Relations
  • Def. 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.
  • Q. Prove that RAA is transitive iff R?R R.
  • Def. 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.

14
7.2 n-ary Relations
  • Def. An n-ary relation R on sets A1,,An,
    written (with signature) RA1An or RA1,,An,
    is simply a subset R ? A1 An.
  • Def. The sets Ai are called the domains of R.
  • The degree of R is n.
  • Def. R is functional in the domain Ai if it
    contains at most one n-tuple (, ai ,) for any
    value ai within domain Ai.

15
Relational Databases
  • Remark.
  • 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

16
Selection Operators
  • Def. 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

17
Selection Operator Example
  • Ex. 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).

18
Projection Operators
  • Def. 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

19
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
20
Join Operator
  • Puts two relations together to form a sort of
    combined relation.
  • Def. 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.

21
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).

22
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.

23
Using Zero-One Matrices
  • Def. To represent a binary relation RAB by an
    AB 0-1 matrix MR mij, let mij 1 iff
    (ai,bj)?R.
  • Ex. Suppose Joe likes Susan and Mary, Fred likes
    Mary, and Mark likes Sally.Then the 0-1 matrix
    representationof the relationLikesBoysGirlsr
    elation is

24
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
25
Using Directed Graphs
  • Def. 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)
26
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
27
7.4 Closures of Relations
  • Def. 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 ? IAThe
    symmetric closure of R is obtained by adding
    (b,a) to R for each (a,b) in R. I.e., it is R ?
    R-1The 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

28
Paths in Digraphs/Binary Relations
  • Def. 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.)
  • Def. 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.

29
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
30
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!

31
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
32
(No Transcript)
33
7.5 Equivalence Relations
  • Def. An equivalence relation (e.r.) on a set A is
    simply any binary relation on A that is
    reflexive, symmetric, and transitive.
  • Ex.
  • 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.

34
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)

35
Equivalence Classes
  • Def. 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.

36
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,

37
Partitions
  • Def. 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.

38
7.6 Partial Orderings
  • Def. 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, ?).

39
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

2
4
8
6
3
1
0
9
5
10
7
40
More to come
  • More slides on partial orderings to be added in
    the future

41
References
  • RosenDiscrete Mathematics and its Applications,
    5eMc GrawHill, 2003
Write a Comment
User Comments (0)
About PowerShow.com