CSE 2353

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CSE 2353 September 25th 2002

• Relations

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Set Partitions

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Math Review

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Hamming DistanceError Correction

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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

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Display of Relations

• X-Y Plot
• Two Lines
• Dia-graph
• Adjacency Matrix

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Types of Relations

• Identity
• Universal
• Inverse
• n-Ary

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Properties of Relations

• Reflexive (a R a)
• Symmetric
• Anti-Symmetric
• Transitive

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Graphic Representation

• Properties of the relation

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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

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Equivalence Relation

• What Properties?
• reflexive?
• anti-symmetric?
• symmetric?
• transitive?

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Equivalence Classes

• Congruence modulo n
• a-b kn

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Partial Ordering

• a R b iff a lt b
• a R b iff a lt b

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Min and Max Elements

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Properties

• Reflexive iff aRa for all a?A
• Symmetric iff aRb -gt bRa for all a,b?A
• Anti-symmetric iff aRb and bRa -gt ab for all
  a,b?A
• Transitive iff aRb and bRc -gt aRc
• Example R is a relation on the real numbers xRy
  iff x ? y