Representing Relations Closures of Relations

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Representing RelationsClosures of Relations
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Learning Objectives

• Understand how to represent relations using
  matrices.
• Understand how to represent relations using
  digraphs.
• Understand what are closures.
• Understand transitive closure and how to
  calculate it.

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Representing Relations

• Connection Matrices
• Let R be a relation from
• A a1 , a2 , . . . , am
• to
• B b1 , b2 , . . . , bn .
• Definition An m x n connection matrix M for R is
• defined by
• Mij 1 if ltai , bj gt is in R,
• 0 otherwise.

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Representing Relations

• Example
• We assume the rows are labeled with the elements
  of A and the columns are labeled with the
  elements of B.
• Let
• A a, b, c
• B e, f, g, h
• R lta,egt, ltc, ggt
• Matrix representation
• Connection matrix M for R
• Note the order of A and B matters

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Representing Relations

• Theorem Let R be a binary relation on a set A
  and let M be its connection matrix. Then
• R is reflexive iff Mii 1 for all i.
• R is symmetric iff M is a symmetric matrix M
  M T
• R is antisymetric if Mij 0 or Mji 0 for all
  i ¹ j.

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Representing Relations

• Combining Connection Matrices
• Definition the join of two matrices M 1 , M 2 ,
  denoted M 1 Ú M 2 , is the component wise boolean
  or of the two matrices.
• Fact If M 1 is the connection matrix for R 1 and
  M 2 is the connection matrix for R 2 then the
  join of M 1 and M 2 , M 1 Ú M 2 is the connection
  matrix for R 1 È R 2.

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Representing Relations

• Definition the meet of two matrices M 1 , M 2 ,
  denoted M 1 Ù M 2 is the component wise boolean
  and of the two matrices.
• Fact If M 1 is the connection matrix for R 1 and
  M 2 is the connection matrix for R 2 then the
  meet of M 1 and M 2 , M 1 Ù M 2 is the connection
  matrix for R 1 Ç R 2 .

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Representing Relations

• Obvious questions
• Given the connection matrix for two relations,
  how does one find the connection matrix for
• The complement?
• The relative complement?
• The symmetric difference?

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Representing Relations

• The Composition
• Definition Let
• M 1 be the connection matrix for R 1 and
• M 2 be the connection matrix for R 2 .
• The boolean product of two connection matrices M
  1 andM 2 , denoted M 1 Ä M 2 , is the connection
  matrix for the composition of R 2 with R 1 , R 2
  o R 1

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Representing Relations

• Why?
• In order for there to be an arc ltx, zgt in the
  composition then there must be and arc ltx, ygt in
  R 1 and an arc lty, in R 2 for some y !
• The Boolean product checks all possible ys. If
  at least
• one such path exists, that is sufficient.
• ____________________
• Note the matrices M 1 and M 2 must be
  conformable the number of columns of M 1 must
  equal the number of rows of M 2 .
• If M 1 is m x n and M 2 is n x p then M 1 Ä M 2
  is m x p.

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Representing Relations

• Example

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Representing Relations

• Note
• there is an arc in R 1 from node 1 in A to node
  2 in B
• there is an arc in R 2 from node 2 in B to node
  2 in
• C.
• Hence there is an arc in R 2 o R 1 from node 1
  in A to
• node 2 in C.
• ___________________
• A useful result

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Representing Relations

• Digraphs
• (see section 6.1)
• Given the digraphs for R1 and R2 , find the
  digraphs for
• R 2 È R 1
• R 2 Ç R 1
• R 2 - R 1

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Representing Relations

• Digraphs
• (see section 6.1)
• Given the digraphs for R1 and R2 , find the
  digraphs for
• R 2 È R 1
• R 2 Ç R 1
• R 2 - R 1
• R 2 Å R 1
• R1c

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Representing Relations

• Digraphs
• (see section 6.1)
• Given the digraphs for R1 and R2 , find the
  digraphs for
• R 2 È R 1
• R 2 Ç R 1
• R 2 - R 1

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Closures of Relations

• Definition The closure of a relation R with
  respect to
• property P is the relation obtained by adding the
  minimum
• number of ordered pairs to R to obtain property
  P.
• In terms of the digraph representation of R
• To find the reflexive closure - add loops.
• To find the symmetric closure - add arcs in the
• opposite direction.
• To find the transitive closure - if there is a
  path from
• a to b, add an arc from a to b.

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Closures of Relations

• Note Reflexive and symmetric closures are easy.
• Transitive closures can be very complicated.
• _________________
• Definition Let A be a set and let D ltx, xgt
  x in A.
• D is called the diagonal relation on A
  (sometimes called the equality relation E).
• Note that D is the smallest (has the fewest
  number of ordered pairs) relation which is
  reflexive on A.

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Closures of Relations

• Reflexive Closure
• Theorem Let R be a relation on A. The reflexive
  closure of R, denoted r(R), is R È D .
• Add loops to all vertices on the digraph
  representation of R.
• Put 1s on the diagonal of the connection
  matrix of R.

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Closures of Relations

• Symmetric Closure
• Definition Let R be a relation on A. Then R -1
  or the
• inverse of R is the relation
• R 1 lt y, x gtlt x, y gtÎ R
• __________________
• Note to get R -1
• reverse all the arcs in the digraph
  representation of R
• take the transpose M T of the connection matrix
  M of R.

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Closures of Relations

• Note This relation is sometimes denoted as R T
  or R c
• and called the converse of R
• The composition of the relation with its inverse
  does not
• necessarily produce the diagonal relation (recall
  that the
• composition of a bijective function with its
  inverse is the
• identity).
• ___________________
• Theorem Let R be a relation on A. The symmetric
• closure of R, denoted s(R ), is the relation R È
  R -1 .

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Closures of Relations

• Examples

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Closures of Relations

• Examples
• If A Z, then r( ¹ ) Z x Z
• If A Z , then s( lt ) ¹.
• What is the (infinite) connection matrix of s(lt)?
• If A Z, then s(?) ?
• Theorem R is symmetric iff R R -1

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Closures of Relations

• Theorem Let R 1 and R 2 be relations from A to
  B. Then

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Closures of Relations

• Paths
• Definition A path of length n in a digraph G is
  a
• sequence of edges ltx 0 , x 1 gtltx 1 , x 2 gt . . .
  ltx n-1 , x n gt.
• The terminal vertex of the previous arc matches
  with the initial vertex of the following arc.
• If x 0 x n the path is called a cycle or
  circuit. Similarly for relations.
• _________________

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Closures of Relations

• Theorem Let R be a relation on A. There is a
  path of
• length n from a to b iff lta, bgt Î R n .
• Proof (by induction)
• Basis An arc from a to b is a path of length 1
• which is in R 1 R. Hence the assertion is true
  for n 1.
• Induction Hypothesis Assume the assertion is
  true
• for n.
• Show it must be true for n1.
• There is a path of length n1 from a to b iff
  there is an x in
• A such that there is a path of length 1 from a to
  x and a
• path of length n from x to b.

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Closures of Relations

• From the Induction Hypothesis,
• lta, xgt Î R
• and since ltx , bgt is a path of length n,
• ltx, bgt Î R n .
• If lta, xgt Î R and ltx, bgt Î Rn ,
• then
• lta, bgt Î Rn o R R n1
• by the inductive definition of the powers of R.
• Q. E. D.
• ______________________

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Closures of Relations

• Useful Results for Transitive Closure
• Theorem
• If A Ì B and C Ì B, then A È C Ì B.
• Theorem
• If R Ì S and T Ì U then R o T Ì S o U.
• Corollary
• If R Ì S then R n Ì S n
• Theorem
• If R is transitive then so is R n

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Closures of Relations

• Transitive Closure
• Recall that the transitive closure of a relation
  R, t(R), is
• the smallest transitive relation containing R.
• Also recall
• R is transitive iff R n is contained in R for all
  n
• Hence, if there is a path from x to y then there
  must be an
• arc from x to y, or ltx, ygt is in R.
• Example
• If A Z and R lt i, i1gt then t(R) lt

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Closures of Relations
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Closures of Relations

• _________________
• Definition The connectivity relation or the star
  closure of the relation R, denoted R, is the set
  of ordered pairs lta, bgt such that there is a path
  (in R) from a to b. Since Rn consists of the
  pairs (a,b) such that there is a path of length n
  from a to b, it follows that

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Closures of Relations

• Examples
• Let A Z and R lti, i1gt. R lt.
• Let A the set of people, R ltx, ygt person
  x is a parent of person y. R ?
• __________________

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Closures of Relations

• Theorem t(R) R.
• Example
• R2
• R3
• R4
• R5
• t(R) R
• ______________________

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Closures of Relations

• Definition The closure of a relation R with
  respect to
• property P is the relation obtained by adding the
  minimum
• number of ordered pairs to R to obtain property
  P.
• In terms of the digraph representation of R
• To find the reflexive closure - add loops.
• To find the symmetric closure - add arcs in the
• opposite direction.
• To find the transitive closure - if there is a
  path from
• a to b, add an arc from a to b.

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Closures of Relations

• Definition The closure of a relation R with
  respect to
• property P is the relation obtained by adding the
  minimum
• number of ordered pairs to R to obtain property
  P.
• In terms of the digraph representation of R
• To find the reflexive closure - add loops.
• To find the symmetric closure - add arcs in the
• opposite direction.
• To find the transitive closure - if there is a
  path from
• a to b, add an arc from a to b.