27-ClosuresEquivRels

27-ClosuresEquivRels - Discussion #27 Chapter 5, Sections...

Info iconThis preview shows pages 1–7. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Discussion #27 Chapter 5, Sections 4.6-7 1/15 Discussion #27 Closures & Equivalence Relations Discussion #27 Chapter 5, Sections 4.6-7 2/15 Topics Reflexive closure Symmetric closure Transitive closure Equivalence relations Partitions Discussion #27 Chapter 5, Sections 4.6-7 3/15 Closure Closure means adding something until done. Normally adding as little as possible until some condition is satisfied Least fixed point similarities Discussion #27 Chapter 5, Sections 4.6-7 4/15 Reflexive Closure Reflexive closure of a relation: R (r) smallest reflexive relation that contains R (i.e. fewest pairs added) R (r) = R I A (R is a relation on a set A, and I A is the identity relation 1s on the diagonal and 0s elsewhere.) 1 1 1 3 1 2 1 1 3 2 1 1 1 1 3 1 1 2 1 1 1 3 2 1 R = R (r) = 2200 x (xRx) Discussion #27 Chapter 5, Sections 4.6-7 5/15 Symmetric Closure Symmetric closure of a relation: R (s) smallest symmetric relation that contains R (i.e. fewest pairs added) R (s) = R R ~ (R ~ is R inverse) 1 1 1 3 1 2 1 1 3 2 1 1 1 3 1 1 2 1 1 3 2 1 R = R ~ = 1 1 1 3 1 1 2 1 1 1 3 2 1 R R ~ = 2200 x 2200 y(xRy yRx) Discussion #27 Chapter 5, Sections 4.6-7 6/15 Transitive Closure Transitive closure of a relation: R (t) = R + smallest transitive relation that contains R (i.e. fewest pairs added) for each path of length i, there must be a direct path. (This follows from x y, y z x z; since, if we also have v x, we must have v z, a path of length 3.) R (t) = R R 2 R 3 R |A| . (No path can be longer than |A|, the number of elements in A.) Discussion #27...
View Full Document

This note was uploaded on 03/02/2012 for the course C S 236 taught by Professor Michaelgoodrich during the Winter '12 term at BYU.

Page1 / 15

27-ClosuresEquivRels - Discussion #27 Chapter 5, Sections...

This preview shows document pages 1 - 7. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online