Equivalence Relations Partial Orderings

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Equivalence RelationsPartial Orderings
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Learning Objectives

• Define what are equivalence relations.
• Define equivalence classes and partition.
• Define what is an equivalence relation induced by
  another relation.
• Define what is a partial order.
• Define what is the least element, greatest
  element, lexicographic order.
• Define Hasse or Poset diagrams.
• Define properties of these diagrams.

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Equivalence Relations

• Definition A relation R on a set A is an
  equivalence relation iff R is
• reflexive
• symmetric
• and
• transitive
• _______________

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Equivalence Relations

• It is easy to recognize equivalence relations
  using
• digraphs.
• The subset of all elements related to a
  particular
• element forms a universal relation (contains all
  possible
• arcs) on that subset. The (sub)digraph
  representing the
• subset is called a complete (sub)digraph. All
  arcs are
• present.
• The number of such subsets is called the rank
  of the
• equivalence relation.

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Equivalence Relations

• Examples
• A has 3 elements

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Equivalence Relations

• Each of the subsets is called an equivalence
  class.
• A bracket around an element means the
  equivalence
• class in which the element lies.
• x y ltx, ygt is in R
• The element in the bracket is called a
  representative
• of the equivalence class. We could have chosen
  any one.
• ____________________
• Examples

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Equivalence Relations

• An interesting counting problem
• Count the number of equivalence relations on a
  set A with
• n elements. Can you find a recurrence relation?
• The answers are
• 1 for n 1
• 3 for n 2
• 5 for n 3
• How many for n 4 ?

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Equivalence Relations

• Definition Let S1, S2, . . ., Sn be a collection
  of subsets of A. Then the collection forms a
  partition of A if the subsets are nonempty,
  disjoint and exhaust A
• Si ¹ Æ
• SiÇSj Æ if i ¹ j
• U Si A

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Equivalence Relations

• Theorem The equivalence classes of an
  equivalence
• relation R partition the set A into disjoint
  nonempty
• subsets whose union is the entire set.
• This partition is denoted A/R and called
• the quotient set, or
• the partition of A induced by R, or,
• A modulo R.
• _______________
• Examples
• A A
• A Æ

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Equivalence Relations

• Theorem Let R be an equivalence relation on A.
  Then either
• a b
• or
• aÇb Æ

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Equivalence Relations

• Theorem If R1 and R2 are equivalence relations
  on A then R1Ç R2 is an equivalence relation on A.
• Proof It suffices to show that the intersection
  of
• reflexive relations is reflexive,
• symmetric relations is symmetric,
• and
• transitive relations is transitive.

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Equivalence 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.
• _________________
• Note Reflexive and symmetric closures are easy.
• Transitive closures can be very complicated.

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Equivalence Relations

• Definition Let R be a relation on A. Then the
  reflexive, symmetric, transitive closure of R,
  tsr(R), is an equivalence relation on A, called
  the equivalence relation induced by R.
• Example
• A aÈb aÈb, c, d
• A/R a, b, c, d

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Equivalence Relations

• Theorem tsr(R) is an equivalence relation
• Proof
• We have to be careful and show that tsr(R) is
  still
• symmetric and reflexive.
• Since we only add arcs vs. deleting arcs when
• computing closures it must be that tsr(R) is
  reflexive since
• all loops ltx, xgt on the diagraph must be present
  when
• constructing r(R).
• If there is an arc ltx, ygt then the symmetric
  closure
• of r(R) ensures there is an arc lty, xgt.

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Equivalence Relations

• Now argue that if we construct the transitive
  closure
• of sr(R) and we add an edge ltx, zgt because there
  is a path
• from x to z, then there must also exist a path
  from z to x
• (why?) and hence we also must add an edge ltz, xgt.
  Hence
• the transitive closure of sr(R) is symmetric.
• Q. E. D.
• _____________________

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Partial Ordering

• Definition Let R be a relation on A. Then R is a
  partial
• order iff R is
• reflexive
• antisymmetric
• and
• transitive
• (A, R) is called a partially ordered set or a
  poset.
• __________________

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Partial Ordering

• Note It is not required that two things be
  related under a partial order. That's the partial
  part of it.
• If two objects are always related in a poset,
  it is called a total order or linear order or
  simple order. In this case (A, R) is called a
  chain.
• _________________

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Partial Ordering

• Examples
• (Z ) is a poset. In this case either a b or
  b a so
• two things are always related. Hence, is a
  total order and
• (Z, ) is a chain.
• If S is a set then (P(S), Í ) is a poset. It
  may not be the case that A Í B or B Í A. Hence, Í
  is not a total order.
• (Z, 'divides') is a poset which is not a
  chain.

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Partial Ordering

• Definition Let R be a total order on A and
  suppose S Í
• A. An element s in S is a least element of S iff
  sRb for
• every b in S.
• Similarly for greatest element.
• Note this implies that lta, sgt is not in R for
  any a unless a
• s. (There is nothing smaller than s under the
  order R).
• ___________________
• Definition A chain (A, R) is well-ordered iff
  every
• subset of A has a least element.
• ___________________

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Partial Ordering

• Examples
• (Z, ) is a chain but not well-ordered. Z does
  not
• have least element.
• (N, ) is well-ordered.
• (N, ³) is not well-ordered.
• __________________

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Partial Ordering

• Lexicographic Order
• Given two posets (A1, R1) and (A2, R2) we
  construct an
• induced partial order R on A1 A2
• lt x1, y1gt R ltx2, y2gt iff
• x1 R1 x2
• or
• x1 x2 and y1 R2 y2.
• _________________

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Partial Ordering

• Example
• Let A1 A2 Z and R1 R2 'divides'.
• Then
• lt2, 4gt R lt2, 8gt since x1 x2 and y1 R2 y2.
• lt2, 4gt is not related under R tolt2, 6gt since x1
  x2
• but 4 does not divide 6.
• lt2, 4gt R lt4, 5gt since x1 R1 x2. (Note that 4 is
  not
• related to 5).
• This definition extends naturally to multiple
  Cartesian
• products of partially ordered sets
• A1 A2 A3 . . . An.

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Partial Ordering

• Example Using the same definitions of Ai and Ri
  as
• above,
• lt 2, 3, 4, 5gt R lt 2, 3, 8, 2gt since x1 x2, y1
  y2
• and 4 divides 8.
• lt2, 3, 4, 5gt is not related to lt3, 6, 8, 10gt
  since 2
• does not divide 3.
• _______________
• Strings
• We apply this ordering to strings of symbols
  where there is
• an underlying 'alphabetical' or partial order
  (which is a
• total order in this case).
• ________________

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Partial Ordering

• Example
• Let A a, b, c and suppose R is the natural
• alphabetical order on A
• a R b and b R c.
• Then
• Any shorter string is related to any longer
  string
• (comes before it in the ordering).
• If two strings have the same length then use
  the
• induced partial order from the alphabetical
  order
• aabc R abac

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Partial Ordering

• Hasse or Poset Diagrams
• To construct a Hasse diagram
• 1) Construct a digraph representation of the
  poset
• (A, R) so that all arcs point up (except the
  loops).
• 2) Eliminate all loops
• 3) Eliminate all arcs that are redundant because
  of
• transitivity
• 4) eliminate the arrows at the ends of arcs since
• everything points up.
• _________________

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Partial Ordering

• The elements of P(a, b, c) are
• Æ, a, b, c, a, b, a, c, b, c, a, b,
  c
• The digraph is

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Partial Ordering

• Maximal and Minimal Elements
• Definition Let (A, R) be a poset. Then a in A is
  a
• minimal element if there does not exist an
  element b in A such that bRa.
• Similarly for a maximal element.
• _________________
• Note there can be more than one minimal and
  maximal
• element in a poset.
• _________________
• Example In the above Hasse diagram, Æ is a
  minimal
• element and a, b, c is a maximal element.

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Partial Ordering

• Least and Greatest Elements
• Definition Let (A, R) be a poset. Then a in A is
  the least
• element if for every element b in A, aRb and b is
  the
• greatest element if for every element a in A,
  aRb.
• Theorem Least and greatest elements are unique.
• Proof
• Assume they are not. . .
• ___________________
• Example
• In the poset above a, b, c is the greatest
  element. Æ is
• the least element.

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Partial Ordering

• Upper and Lower Bounds
• Definition Let S be a subset of A in the poset
  (A, R). If
• there exists an element a in A such that sRa for
  all s in S,
• then a is called an upper bound. Similarly for
  lower
• bounds.
• Note to be an upper bound you must be related to
  every
• element in the set. Similarly for lower bounds.
• _________________
• Example
• In the poset above, a, b, c, is an upper
  bound for
• all other subsets. Æ is a lower bound for all
  other subsets.

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Partial Ordering

• Least Upper and Greatest Lower Bounds
• Definition If a is an upper bound for S which is
  related
• to all other upper bounds then it is the least
  upper bound,
• denoted lub(S). Similarly for the greatest lower
  bound,
• glb(S).
• ___________________
• Example
• Consider the element a.
• Since a, b, c, a, b a, c and a
• are upper bounds and a is related to all of
  them, a
• must be the lub. It is also the glb.

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Partial Ordering

• Lattices
• Definition A poset is a lattice if every pair of
  elements has a lub and a glb.
• __________________
• Examples
• In the poset (P(S), Í), lub(A, B) A È B. What
  is
• the glb(A, B)?

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Partial Ordering

• Consider the elements 1 and 3.
• Upper bounds of 1 are 1, 2, 4 and 5.
• Upper bounds of 3 are 3, 2, 4 and 5.
• 2, 4 and 5 are upper bounds for the pair 1 and
  3.
• There is no lub since
• - 2 is not related to 4
• - 4 is not related to 2
• - 2 and 4 are both related to 5.
• There is no glb either.
• The poset is not a lattice.

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Partial Ordering

• Topological Sorting
• We impose a total ordering R on a poset
  compatible with the partial order.
• Useful in PERT charts to determine an ordering
  of tasks
• Useful in rendering in graphics to render
  objects from back to front to obscure hidden
  surfaces
• A painter uses a topological sort when applying
  paint to a canvas - he/she paints parts of the
  scene furthest from the view first.
• Algorithm To sort a poset (S, R).
• Select a (any) minimal element and put it in
  the list.
• Delete it from S.
• Continue until all elements appear in the list
  (and S
• is void).

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Partial Ordering

• Example
• Consider the rectangles T and the relation R
  is more distant than. Then R is a partial order
  on the set of rectangles.
• Two rectangles, Ti and Tj, are related, Ti R Tj,
  if Ti is more distant from the viewer than Tj.

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Partial Ordering

• Then 1R2, 1R4, 1R3, 4R9, 4R5, 3R2, 3R9, 3R6, 8R7.
• The Hasse diagram for R is

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Partial Ordering

• Draw 1 (or 8) and delete 1 from the diagram to
  getNow draw 4 (or 3 or 8) and delete from
  the diagram.
• Always choose a minimal element. Any one will
  do.
• ...and so forth.