3.8AABC030109

# 3.8AABC030109 - 1 3.8 AABC Recall the reduced AA table from...

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1 3.8. AABC. Recall the reduced AA table from section 3.4. REDUCED AA 1. B . fA A . gA. ¬ INF. AL. ¬ ALF. ¬ FIN. NON. 2. B . fA A . gB. ¬ INF. AL. ¬ ALF. ¬ FIN. NON. 3. B . fA A . gC. ¬ INF. AL. ¬ ALF. ¬ FIN. NON. 4. C . fA A . gA. ¬ INF. AL. ¬ ALF. ¬ FIN. NON. 5. C . fA A . gB. ¬ INF. AL. ¬ ALF. ¬ FIN. NON. 6. C . fA A . gC. ¬ INF. AL. ¬ ALF. ¬ FIN. NON. Recall the reduced AB table from section 3.5. REDUCED AB 1. A . fA B . gA. INF. AL. ALF. FIN. NON. 2. A . fA B . gB. INF. AL. ALF. FIN. NON. 3. A . fA B . gC. INF. AL. ALF. FIN. NON. 4. C . fA B . gA. INF. AL. ALF. FIN. NON. 5. C . fA B . gB. INF. AL. ALF. FIN. NON. 6. C . fA B . gC. INF. AL. ALF. FIN. NON. The reduced BC table is obtained from the reduced AB table via the permutation sending A to B, B to C, C to A. We use 1'-6' to avoid any confusion. REDUCED BC 1’. B . fB C . gB. INF. AL. ALF. FIN. NON. 2’. B . fB C . gC. INF. AL. ALF. FIN. NON. 3’. B . fB C . gA. INF. AL. ALF. FIN. NON. 4’. A . fB C . gB. INF. AL. ALF. FIN. NON. 5’. A . fB C . gC. INF. AL. ALF. FIN. NON. 6’. A . fB C . gA. INF. AL. ALF. FIN. NON. All attributes are determined from the reduced AA table, except for AL and NON. So we merely have to determine the status of AL and NON. part 1. B . fA A . gA. 1,1’. B . fA A . gA, B . fB C . gB. ¬ INF. AL. ¬ ALF. ¬ FIN. NON. 1,2’. B . fA A . gA, B . fB C . gC. ¬ INF. AL. ¬ ALF. ¬ FIN. NON.

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2 1,3’. B . fA A . gA, B . fB C . gA. ¬ INF. AL. ¬ ALF. ¬ FIN. NON. 1,4’. B . fA A . gA, A . fB C . gB. ¬ INF. ¬ AL. ¬ ALF. ¬ FIN. ¬ NON. 1,5’. B . fA A . gA, A . fB C . gC. ¬ INF. ¬ AL. ¬ ALF. ¬ FIN. ¬ NON. 1,6’. B . fA A . gA, A . fB C . gA. ¬ INF. ¬ AL. ¬ ALF. ¬ FIN. ¬ NON. The following pertains to 1,1’, 1,3’. LEMMA 3.8.1. B . fA A . gA, B . fB C . gX has AL, provided X {A,B}. Proof: Let f,g ELG(N) and p > 0. Let B = [n,n+p], where n is sufficiently large. By Lemma 3.3.3, let A be unique such that A [n, ) A . gA. Let C = [n, )\gX. Note that B fA = B fB = B gA = A gA = B gB = C gX = . Hence B A,C. Also B fA [n, ) = A gA, and B fB [n, ) C gX. QED The following pertains to 1,2’. LEMMA 3.8.2. B . fA A . gA, B . fB C . gC has AL. Proof: Let f,g
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3.8AABC030109 - 1 3.8 AABC Recall the reduced AA table from...

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