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Partitions of the set N

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September 1, 2010
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Problem Statement

(a) Show that the set N\mathbb N of all positive integers can be partitioned into three disjoint subsets A,BA, B, and CC satisfying the following conditions: A2=A,B2=C,C2=B,A^2 = A, B^2 = C, C^2 = B, AB=B,AC=C,BC=A,AB = B, AC = C, BC = A, where HKHK stands for {hk∣h∈H,k∈K}\{hk | h \in H, k \in K\} for any two subsets H,KH, K of N\mathbb N, and H2H^2 denotes HH.HH. (b) Show that for every such partition of N\mathbb N, min⁡{n∈N∣n∈A and n+1∈A}\min\{n \in N | n \in A \text{ and } n + 1 \in A\} is less than or equal to 77.77.
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