Math 630 — Fall 2010
Quiz 3
Due Monday, October 25, in class.
Problem 1.
P
Let S be the family of k-subsets of [n], where n is even. For A ∈ S, define w(A) = i∈A i, and
set S + = {A ∈ S | w(A) is even}, S − = {A ∈ S | w(A) is odd}. Use an involution to prove that
+
−
#S − #S =
0
−1k/2
n/2
k/2
k odd,
k even.
Note that this number may be negative. What does that tell you about the involution you need?
Problem 2.
Let L be a finite semimodular lattice. Show that the following are equivalent.
1. L is relatively complemented.
2. L is atomic.