Equivalence uandvarestrongly Relations connected. u G * * v AND v - PowerPoint PPT Presentation

�������� Mathematics for Computer Science two-way walks MIT 6.042J/18.062J walk from u to v and back from v to u: Equivalence u and v are strongly Relations connected. u G * * v AND v G * * u AND Albert R Meyer March 22, 2013 Albert R Meyer March 22, 2013 equiv.1 equiv.2 symmetry equivalence relations transitive, relation R on set A symmetric & is symmetric iff y reflexive a R b IMPLIES b R a a R b IMPLIES b R a R b IMPLIES b R a Albert R Meyer March 22, 2013 Albert R Meyer March 22, 2013 equiv.3 equiv.4 ��

�������� equivalence relations equivalence relations examples: Theorem: R is an equiv rel iff • = (equality) R is the strongly • - (mod n) connected relation • same size of some digraph • same color Albert R Meyer March 22, 2013 Albert R Meyer March 22, 2013 equiv.5 equiv.6 Graphical Properties of Relations Reflexive Asymmetric Representing NO Equivalences Symmetric Transitive Albert R Meyer March 22, 2013 Albert R Meyer March 22, 2013 equiv.7 equiv.8 ��

�������� Representing equivalences Representing equivalences Theorem: For total function f:A � B Relation R on set A is define relation ≡ f on A: an equiv. relation IFF a ≡ f a’ IFF f(a) = f(a’) � R is ≡ f � for some f:A � B Albert R Meyer March 22, 2013 Albert R Meyer March 22, 2013 equiv.9 equiv.10 representing ≡ (mod n) Representing equivalences For partition ∏ of A ≡ (mod n) is define relation ≡ ∏ on A: ≡ f where a ≡ ∏ a’ IFF a, a’ are in f(k) ::= rem(k,n) the same block of ∏ � Albert R Meyer March 22, 2013 Albert R Meyer March 22, 2013 equiv.11 equiv.12 ��

�������� Representing equivalences Theorem: Relation R on set A is an equiv. relation IFF R is ≡ ∏ for some partition ∏ of A Albert R Meyer March 22, 2013 equiv.13 ��

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