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.