CSE 2353 – September 25th 2002

Relations

Set Partitions

Math Review

Hamming DistanceError 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

• Two Lines

• Dia-graph

• “Adjacency” Matrix

Types of Relations

• Identity

• Universal

• Inverse

• n-Ary

Properties of Relations

• Reflexive (a R a)

• Symmetric

• Anti-Symmetric

• Transitive

Graphic Representation

• Properties of the relation:

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

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

Equivalence Classes

• Congruence modulo n – a-b = kn

Partial Ordering

• a R b iff a <= b

• a R b iff a < b

Min and Max Elements

Properties

• Reflexive iff aRa for all aA

• Symmetric iff aRb -> bRa for all a,bA

• Anti-symmetric iff aRb and bRa -> a=b for all a,bA

• Transitive iff aRb and bRc -> aRc

• Example: R is a relation on the real numbers: xRy iff x y