lec0427 - CS 173: Discrete Mathematical Structures Cinda...

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CS 173: Discrete Mathematical Structures Cinda Heeren heeren@cs.uiuc.edu Siebel Center, rm 2213 Office Hours: BY APPOINTMENT
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Cs173 - Spring 2004 CS 173 Announcements Hwk #12 available, due 4/30, 8a Final Exam: 5/10, 7-10p, Siebel 1404 Email Cinda with conflict. Problem #13 available, due 5/3, 8a.
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Cs173 - Spring 2004 CS173 Partially Ordered Sets (POSets) Ex. A common partial order on bit strings of length n, {0,1} n , is defined as: a 1 a 2 …a n b 1 b 2 …b n If and only if a i b i , 2200 i. 0110 and 1000 are “incomparable” … We can’t tell which is “bigger.” As a bit of an aside, this relation is exactly the same as the last example, (2 S ). A. 0110 1000 B. 0000 C. 1110 D. 10111 Huh? Set S, on which we build 2 S , has a size. That’s n. Suppose S is {a,b}. Then 2 S = { {}, {a}, {b}, {a,b} } Think of bit strings as membership indicators for the elts of S S can be represented by {00,10,01,11}
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Cs173 - Spring 2004 CS173 Partially Ordered Sets (POSets) 0110 and 1000 are “incomparable” … We can’t tell which is “bigger.” As a bit of an aside, this relation is exactly the same as the last example, (2 S , ). In the string relation, we said 00 01 because every bit in 00 is less than or = the corresp bit in 01. String on the right has at least all the 1 bits of the left, maybe more. If each 1 represents an element in S, then right side has all elts of the left, maybe more. Set S, on which we build 2 S , has a size. That’s n. Suppose S is {a,b}. Then 2 S = { {}, {a}, {b}, {a,b} } Think of bit strings as membership indicators for the elts of S S can be represented by {00,10,01,11}
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Cs173 - Spring 2004 CS173 Partially Ordered Sets (POSets) Let (S, ) be a PO. If a b, or b a, then a and b are comparable. Otherwise, they are incomparable. Ex. In poset (Z + , | ), 3 and 6 are comparable, 6 and 3 are comparable, 3 and 5 are not, 8 and 12 are not.
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lec0427 - CS 173: Discrete Mathematical Structures Cinda...

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