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

u G*

v AND v G* u

a R b IMPLIES b R a

• equiv.1

Albert R Meyer March 22, 2013

Mathematics for Computer Science

MIT 6.042J/18.062J

Equivalence Relations

equiv.2 Albert R Meyer March 22, 2013

two-way walks walk from u to v and back from v to u: u and v are strongly connected.

*

AND

*

equiv.3 Albert R Meyer March 22, 2013

symmetry relation R on set A is symmetric iff y a R b IMPLIES b R a

equivalence relations

transitive, symmetric & reflexive

Albert R Meyer March 22, 2013 equiv.4

R b IMPLIES b R a

• equiv.5

Albert R Meyer March 22, 2013

Theorem:

R is an equiv rel iff R is the strongly connected relation

• f some digraph

equivalence relations

equiv.6 Albert R Meyer March 22, 2013

examples:

• = (equality)
• - (mod n)
• same size
• same color

equivalence relations

equiv.7 Albert R Meyer March 22, 2013

Graphical Properties of Relations Reflexive Transitive Symmetric Asymmetric NO

Representing Equivalences

Albert R Meyer March 22, 2013 equiv.8

• equiv.9

Albert R Meyer March 22, 2013

Representing equivalences

For total function f:AB define relation ≡f on A:

a ≡f a’ IFF f(a) = f(a’)

equiv.10 Albert R Meyer March 22, 2013

Representing equivalences

Theorem:

Relation R on set A is an equiv. relation IFF

R is ≡f

for some f:AB

equiv.11 Albert R Meyer March 22, 2013

representing ≡ (mod n)

≡ (mod n) is

≡f where f(k) ::= rem(k,n)

equiv.12 Albert R Meyer March 22, 2013

Representing equivalences

For partition ∏ of A define relation ≡∏ on A:

a ≡∏ a’ IFF a, a’ are in the same block of ∏

• equiv.13

Albert R Meyer March 22, 2013

Representing equivalences

Theorem:

Relation R on set A is an

• equiv. relation IFF

R is ≡∏

for some partition ∏ of A

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