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Equivalence Relations. Partial Ordering Relations

Equivalence Relations. Partial Ordering Relations. 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.

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Equivalence Relations. Partial Ordering Relations

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  1. Equivalence Relations.Partial Ordering Relations

  2. 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 xRy mean that x and y lie in the same member of P.

  3. 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, • Then P= {A, B, C} is a partition of S={1,2,3,4,5,6}. • Then we can establish a relation “xRy 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)}

  4. 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.

  5. Antisymmetric Relation • A relation R on a set S is called antisymmetric if, whenever xRy and yRx 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.

  6. 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, xRx is true . • R is antisymmetric, that is . • R is transitive, that is .

  7. 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 define ARB if . R is reflexive ( ), antisymmetric and transitive . Thus R is a partial order.

  8. 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 xprecedesto vector y if for all n components of these two vectors the following property holds . For example, if n=3: , but . • The relation is a partial order on the set of n-dimensional binary vectors.

  9. 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.

  10. 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

  11. 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.

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

  13. 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.

  14. 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 yRx is x itself: • is called a maximal element of S with respect to R if the only element satisfying xRy is x itself:

  15. 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)

  16. 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, xRx 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 xRy means that x takes the same class as y. R is a tolerance: it is reflexive, symmetric, but not transitive.

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