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Mathematics 501 Homework (due Oct 11) A. Hulpke 30) For a natural number n, denote by D(n) the lattice of divisors of n. Show that if n = p1e1 · · · pakk with pi 6= p j primes, then D(n) is (isomorphic to) the direct product of the lattices D(p1e1 ),. . . ,D(pekk ). 31) The book defines on p. 201 the Möbius function by the property ∑ µ(x, z) = δx, y x≤ z≤ y (defining as usual δx, y to be 1 whenever x = y and 0 otherwise) but uses later that ∑ µ(z, y) = x≤ z≤ y δx, y . Why does this identity hold (and why could one call it the definition)? 32) Let a,b be elements of a poset P. Prove that µ(a, b) = ∑(−1)i ci , i≥0 where ci is the number of chains a = x0 < x1 < · · · < xi = b. (Hint: Show that the right hand side fulfills the defining property.) 33) Show that the principle of inclusion and exclusion is a special case of the Möbius inversion formula, using the poset of subsets. 34) Let { A1 , . . . , An } be a family of sets with an SDR and x ∈ A1 . Show that there always is an SDR containing x, but (by giving a counterexample) that it is not necessarily possible to find an SDR in which x represents A1 . 35∗ ) (Due to P ETER B RO M ILTERSEN) The names of 100 prisoners are placed in 100 closed boxes, one name per box. The boxes are lined up on a table in a room. One by one the prisoners are led to the room. Each prisoner may look into 50 boxes, but must leave the room exactly as he found it. He may not communicate with the other prisoners about his findings. Unless every prisoner opens a box containing his own name (the probability of this being 2−100 if every prisoner opens boxes by random !) all prisoners will have to triple-grade Calculus 1 exams for the next month. The prisoners however have the possibility to plot a strategy in advance. Find a strategy that has a success probability of over 30%. Hint: Problem 28 Problems marked with ∗ are bonus problems for extra credit.