Cse 2353 september 25 th 2002
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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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Cse 2353 september 25 th 2002

CSE 2353 – September 25th 2002

Relations


Set partitions

Set Partitions


Math review

Math Review


Hamming distance error correction

Hamming DistanceError Correction


Relations

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

Display of Relations

  • X-Y Plot

  • Two Lines

  • Dia-graph

  • “Adjacency” Matrix


Types of relations

Types of Relations

  • Identity

  • Universal

  • Inverse

  • n-Ary


Properties of relations

Properties of Relations

  • Reflexive (a R a)

  • Symmetric

  • Anti-Symmetric

  • Transitive


Graphic representation

Graphic Representation

  • Properties of the relation:


Set terms

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


Equivalence relation

Equivalence Relation

  • What Properties?

    • reflexive?

    • anti-symmetric?

    • symmetric?

    • transitive?


Equivalence classes

Equivalence Classes

  • Congruence modulo n

    • a-b = kn


Partial ordering

Partial Ordering

  • a R b iff a <= b

  • a R b iff a < b


Min and max elements

Min and Max Elements


Properties

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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