27-ClosuresEquivRels

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

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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...
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## 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.

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

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