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# lec0420 - CS 173 Discrete Mathematical Structures Cinda...

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CS 173: Discrete Mathematical Structures Cinda Heeren [email protected] Siebel Center, rm 2213 Office Hours: BY APPOINTMENT

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Cs173 - Spring 2004 CS 173 Announcements Hwk #11 available, due 4/23, 8a Final Exam:  5/10, 7-10p, Siebel 1404 Problem #13 available today.
Cs173 - Spring 2004 CS173 Properties of Relations - techniques… Let R be a relation on positive integers, R={(x,y): 3|(x-y)} Yes Is R transitive? Suppose (x,y) and (y,z) are in R. Then we can write 3j = (x-y) and 3k = (y-z) Definition of  “divides” Can we say 3m = (x-z)?  Is (x,z) in R? Add prev eqn to get: 3j + 3k = (x-y) + (y-z)  3(j + k) = (x-z)

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Cs173 - Spring 2004 CS173 Properties of Relations - techniques… Let R be a relation on positive integers, R={(x,y): 3|(x-y)} Yes Is R transitive? Is it reflexive? Yes Is (x,x) in R, for all x? Does 3k = (x-x) for some k? Definition of  “divides” Yes, for k=0.
Cs173 - Spring 2004 CS173 Properties of Relations - techniques… Let R be a relation on positive integers, R={(x,y): 3|(x-y)} Yes Is R transitive? Is it reflexive? Yes Is it symmetric? Yes Suppose (x,y) is in R. Then 3j = (x-y) for some j. Definition of  “divides” Yes, for k=-j. Does 3k = (y-x) for some k?

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Cs173 - Spring 2004 CS173 Properties of Relations - techniques… Let R be a relation on positive integers, R={(x,y): 3|(x-y)} Yes Is R transitive? Is it reflexive? Yes Is it symmetric? Yes Is it anti-symmetric? No Suppose (x,y) is in R. Then 3j = (x-y) for some j. Definition of  “divides” Yes, for k=-j. Does 3k = (y-x) for some k?
Cs173 - Spring 2004 CS173 More than one relation Suppose we have 2 relations, R 1  and R 2 , and recall that relations are just sets!  So we can take unions, intersections, complements,  symmetric differences, etc. There are other things we can do as well…

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Cs173 - Spring 2004 CS173 More than one relation Let R be a relation from A to B (R   AxB), and let S be a relation from B to C  (S   BxC).  The composition of R and S is the relation from A to C (S °   AxC): S ° R = {(a,c):  5  b B, (a,b)   R, (b,c)   S} S ° R = {(1,u),(1,v),(2,t),(3,t),(4,u)} A B C 1 2 3 4 x y z s t u v R S
Cs173 - Spring 2004 CS173 More than one relation Let R be a relation on A. Inductively define R 1  = R R n+1  = R n   °  R R 2  = R 1 ° R = {(1,1),(1,2),(1,3),(2,3),(3,3),(4,1), (4,2)} A A A 1 2 3 4 1 2 3 4 R R 1 1 2 3 4

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Cs173 - Spring 2004 CS173 More than one relation Let R be a relation on A. Inductively define R 1  = R R n+1  = R n   °  R R 3  = R 2 ° R = {(1,1),(1,2),(1,3),(2,3),(3,3),(4,1),(4,2),(4,3)} A A A 1 2 3 4 1 2 3 4 R R 2 1 2 3 4 … = R4  = R5  = R6…
Cs173 - Spring 2004 CS173 Relations - A Theorem: If R is a transitive relation, then R n    R,  2200 n.

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lec0420 - CS 173 Discrete Mathematical Structures Cinda...

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