Download MATH 4707 PROBLEM SET 2 1. Required problems

MATH 4707 PROBLEM SET 2
FINAL VERSION; DUE WEDNESDAY 10/1
1. Required problems
• Exercises from the text: 2.5.5, 2.5.8, 3.2.2(a–c), 3.2.3 (in parts b and c, n = 8), 3.6.4,
3.8.5, 3.8.6, 3.8.12, 3.8.13, 4.1.3, 4.2.2, 4.2.4, 4.2.6
• Let Mn be the (n + 1) × (n + 1) matrix with rows
and columns numbered 0, 1, . . . , n
i
such that the entry in row i and column j is j . Compute the inverse matrices of
M0 , M1 , M2 and M3 . (No need to do this by hand.) What do you notice? (On this
exercise only, no explanation required. But, you might reflect on the relationship
with Inclusion-Exclusion.)
• Exercise 4 from Dennis White’s handout
2. Optional problems
• Draw the first 4, 8 or 16 rows of Pascal’s triangle. Mark those entries that are even
in one color and those that are odd in another color. What pattern do you observe?
(Note: since knowing the parity of two numbers is enough information to know the
parity of their sum, you can do all of the arithmetic modulo 2, just keeping track of
the parity of the entry in each row to get the parity of the entries in the next row.)
• Exercises from the text: 2.5.7, 3.6.3, 3.7.1, 3.8.8, 3.8.9, 3.8.11, 3.8.14, 4.2.5, 4.3.4,
4.3.7
• Exercises 1, 2, 3 and 5 from Dennis White’s handout
• Suppose that I tell you that a sequence (an ) satisfies a0 = 0 and a9 = 34 and
an+1 = an + an−1 for n ≥ 1. Does this uniquely define the sequence?
• Putnam 1985 A1: Determine, with proof, the number of ordered triples (A1 , A2 , A3 )
of sets which have the property that A1 ∪A2 ∪A3 = {1, 2, . . . , 10} and A1 ∩A2 ∩A3 = ∅.
• Putnam 1993 A3: Let Pn be the set of subsets of {1, 2, . . . , n}. Let c(n, m) be the
number of functions f : Pn → {1, 2, . . . , m} such that f (A ∩ B) = min(f (A), f (B)).
Prove that
m
X
c(n, m) =
j n.
j=1
• Putnam 1990 A6: Call an ordered pair (S, T ) of subsets of {1, 2, . . . , n} admissible
if s > |T | for each s ∈ S and t > |S| for each t ∈ T . How many admissible ordered
pairs of subsets of {1, 2, . . . , 10} are there? (Prove your answer.)
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