Transcript Basic Principles - Texas A&M University

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 S={1,2,3,4,5,6}. Let A={1,3,4}, B={2,6}, and
C={5}. Let some equivalence relation is defined
on these sets. Evidently,
A  B  C  S; A  B  A  C  B  C  
• Then P= {A, B, C} is a partition of
S={1,2,3,4,5,6}.
• 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
x=y.
• Examples. Relations “≤” and “≥” on the set Z
of integer numbers. If x ≤ y and y ≤ x then
always x=y. If x ≥ y and y ≥ x then always x=y.
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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:
1) R is reflexive, that is, x R x is true x  X .
2) R is antisymmetric, that is x R y; y R x  x  y .
3) R is transitive, that is x R y; y R z  x R z .
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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 S={A,B,C,…} be a set whose elements are
other sets. For A, B  S define A R B if A  B .
R is reflexive ( A  A ),
antisymmetric A  B, B  A  A  B
and transitive A  B, B  C  A  C .
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 x y
if for all n components of these two vectors
the following property holds xi  yi , i  1,..., n .
For example, if n=3: 0,0,0 0,0,1 1,0,1 1,1,1 ,
but  0,1,1 (1,0,0);  0,1,0 (1,0,1).
• The relation " " is a partial order on the set
of n-dimensional binary vectors.
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Partial Ordering Relations. Examples
• Let S={A, 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 a1, b1  S1; a2 , b2  S2 .
Then  a1, a2  R b1, b2  if and only if one of the
following is true: (1) a1  b1 &  a1 R1 b1 
(2) a1  b1 &  a2 R2 b2 
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
n
E

0,1
E

E

...

E

E
• If 2   then 2 2
2
2 is a set of
n times
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
x, y  S : x R y or y R x
• 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.
• x  S is called a minimal element of S with
respect to R if the only element y  S
satisfying y R x is x itself: y R x  y  x
• x  S is called a maximal element of S with
respect to R if the only element y  S
satisfying x R y is x itself: x R y  y  x
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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 n=3: (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:
1) R is reflexive, that is, x R x is true x  X
2) R is symmetric, that is x R y  y R x
• 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.
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