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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.