1. Equivalence Relations. Partial Ordering Relations 1

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

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 “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)} 3

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. 4

5. 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. 5

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: 1) R is reflexive, that is, x R x is true. 2) R is antisymmetric, that is. 3) R is transitive, that is. 6

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

8. Partial Ordering Relations. Examples Let us consider a set of n-dimensional binary vectors E 2 n ={(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 n=3:, but. The relation is a partial order on the set of n-dimensional binary vectors. 8

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. 9

10. Lexicographic Order If R 1 is a partial order on set S 1 and R 2 is a partial order on set S 2 then we can define the following relation R on the Cartesian product S 1 xS 2. Let. Then if and only if one of the following is true: R is called the lexicographic order on S 1 xS 2 10

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 R 1 is a partial order on set S 1 and R 2 is a partial order on set S 2 then the lexicographic order is a partial order on the Cartesian product S 1 xS 2. 11

12. Lexicographic Order If then is a set of n- dimensional binary vectors. The relation establishes a lexicographic order on E 2 n 12

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. 13

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 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: 14

15. Minimal and Maximal Elements In the set E 2 n 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) 15

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: 1) R is reflexive, that is, x R x is true 2) 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. 16