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CSE 2353 – September 25 th 2002

CSE 2353 – September 25 th 2002. Relations. Set Partitions. Math Review. Hamming Distance Error Correction. Relations. A R B is a subset of A X B a  A is related to b  B iff (a,b)  R Example: A = B = {1,2,3,4,5,6}; R = {(a,b): a divides b}. Display of Relations. X-Y Plot

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CSE 2353 – September 25 th 2002

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  1. CSE 2353 – September 25th 2002 Relations

  2. Set Partitions

  3. Math Review

  4. Hamming DistanceError Correction

  5. Relations • A R B is a subset of A X B • a  A is related to b  B iff (a,b)  R • Example: • A = B = {1,2,3,4,5,6}; • R = {(a,b): a divides b}

  6. Display of Relations • X-Y Plot • Two Lines • Dia-graph • “Adjacency” Matrix

  7. Types of Relations • Identity • Universal • Inverse • n-Ary

  8. Properties of Relations • Reflexive (a R a) • Symmetric • Anti-Symmetric • Transitive

  9. Graphic Representation • Properties of the relation:

  10. Set Terms • R  S • R  S • R and S are Reflexive • R and S are Symmetric • R and S are anti-symmetric • R and S are Transitive

  11. Equivalence Relation • What Properties? • reflexive? • anti-symmetric? • symmetric? • transitive?

  12. Equivalence Classes • Congruence modulo n • a-b = kn

  13. Partial Ordering • a R b iff a <= b • a R b iff a < b

  14. Min and Max Elements

  15. Properties • Reflexive iff aRa for all aA • Symmetric iff aRb -> bRa for all a,bA • Anti-symmetric iff aRb and bRa -> a=b for all a,bA • Transitive iff aRb and bRc -> aRc • Example: R is a relation on the real numbers: xRy iff x  y

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