Equivalence Relations. Partial Ordering Relations

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Equivalence Relations.Partial Ordering Relations
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Equivalence Relation and Partition

• Every equivalence relation on S gives rise to a
  partition of S by taking the family of subsets in
  the partition to be the equivalence classes of
  the equivalence relation.
• If P is a partition of S, we can define a
  relation R on S by letting x R y mean that x and
  y lie in the same member of P.

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Equivalence Relation and Partition

• Let S1,2,3,4,5,6. Let A1,3,4, B2,6, and
  C5. Let some equivalence relation is defined
  on these sets. Evidently,
• Then P A, B, C is a partition of
  S1,2,3,4,5,6.
• Then we can establish a relation x R y means
  that x and y lie in the same member of P
• R(1,1),(1,3),(1,4),(3,1),(3,3),(3,4),(4,1),(4,3
  ), (4,4), (2,2), (2,6), (6,2), (6,6), (5,5)

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Equivalence Relation and Partition

• Theorem.
• An equivalence relation R on S gives rise to a
  partition P of S, in which the members of P are
  the equivalence classes of R.
• A partition P of S induces an equivalence
  relation R in which any two elements x and y are
  related by R whenever they lie in the same member
  of P. Moreover, the equivalence classes of this
  relation are members of P.

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Antisymmetric Relation

• A relation R on a set S is called antisymmetric
  if, whenever x R y and y R x are both true, then
  xy.
• Examples. Relations and on the set Z of
  integer numbers. If x y and y x then always
  xy. If x y and y x then always xy.

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Partial Ordering Relations

• A relation R on a set S is called a partial
  ordering relation, or simply a partial order, if
  the following 3 properties hold for this
  relation
• R is reflexive, that is, x R x is true
  .
• R is antisymmetric, that is
  .
• R is transitive, that is
  .

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Partial Ordering Relations. Examples

• Relations and are partial orders on sets
  Z of integer numbers and R of real numbers.
• Let SA,B,C, be a set whose elements are other
  sets. For define A R B if
  . R is reflexive ( ),
• antisymmetric
• and transitive
  .
• Thus R is a partial order.

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Partial Ordering Relations. Examples

• Let us consider a set of n-dimensional binary
  vectors E2n(0,,0), (0,,01),,(1,,1). We
  say that vector x precedes to vector y
  if for all n components of these two vectors
  the following property holds
  . For example, if
  n3
  , but
  .
• The relation is a partial order on the
  set of n-dimensional binary vectors.

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Partial Ordering Relations. Examples

• Let SA, B, C, D, E, F, G be a set of classes
  from some program curriculum. Let us define
  relation as follows x is related to y if class x
  is an immediate prerequisite for class y.
• This relation is a partial order.

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Lexicographic Order

• If R1 is a partial order on set S1 and R2 is a
  partial order on set S2 then we can define the
  following relation R on the Cartesian product
  S1xS2. Let .
• Then if and only
  if one of the following is true
• R is called the lexicographic order on S1xS2

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Lexicographic Order

• The lexicographic order is also referred to as a
  dictionary order, because it corresponds to
  the sequence in which words are listed in a
  dictionary.
• Theorem. If R1 is a partial order on set S1 and
  R2 is a partial order on set S2 then the
  lexicographic order is a partial order on the
  Cartesian product S1xS2.

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Lexicographic Order

• If then
  is a set of
• n- dimensional binary vectors . The relation
  establishes a lexicographic order on E2n

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Total (Linear) Order

• A partial order R on set S is called a total
  order (or a linear order) on S if every pair of
  elements in S can be compared, that is
• Relations and are total orders on sets Z
  of integer numbers and R of real numbers.
• Relation is not a total order.

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Minimal and Maximal Elements

• Let R be a partial order on set S.
• is called a minimal element of S with
  respect to R if the only element
  satisfying y R x is x itself
• is called a maximal element of S with
  respect to R if the only element
  satisfying x R y is x itself

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Minimal and Maximal Elements

• In the set E2n of n-dimensional binary vectors
  (0,,0) is a minimal element and (1,,1) is a
  maximal element.
• For n3 (0,0,0), (0,0,1), (0,1,0), (0,1,1),
  (1,0,0), (1,0,1), (1,1,0), (1,1,1)

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Tolerance Relation

• A relation R on a set S is called a tolerance
  relation, or simply a tolerance, if the following
  2 properties hold for this relation
• R is reflexive, that is, x R x is true
  
• R is symmetric, that is
  
• Thus, a tolerance is not transitive.
• Example. Let S be the set of all students in some
  university. Let x R y means that x takes the same
  class as y. R is a tolerance it is reflexive,
  symmetric, but not transitive.