6.3Refute071410

6.3Refute071410 - 1 6.3 A Refutation In Proposition A can...

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1 6.3. A Refutation. In Proposition A, can we replace ELG by the simpler and more basic SD? We refute this in a strong way. In particular, we refute Proposition C with ELG removed. PROPOSITION α . For all f,g SD BAF there exist A,B,C INF such that A . fA C . gB A . fB C . gC. We will even refute the following weaker Proposition. PROPOSITION β . Let f,g SD BAF. There exist A,B,C N, |A| 4, such that A . fA C . gB A . fB C . gC. We assume Proposition β , and derive a contradiction. We begin with a modification of Lemmas 5.1.6 and 5.1.7. Basically, these go through without any change in the proof, but we provide some additional details. LEMMA 5.1.6'. Let f,g SD BAF. There exist f',g' SD BAF such that the following holds. i) g'S = g(S*) 6S+2; ii) f'S = f(S*) g'S 6f(S*)+2 2S*+1 3S*+1. Proof: In the proof of Lemma 5.1.6, f',g' are constructed explicitly from f,g. It is obvious that if f,g SD BAF, then f',g' SD BAF. The verification goes through without change. QED LEMMA 5.1.7'. Let f,g SD BAF and rng(g) 6N. There exist A B C N\{0}, |A| 3, such that i) fA 6N B gB; ii) fB 6N C gC; iii) fA 2N+1 B; iv) fA 3N+1 B; v) fB 2N+1 C; vi) fB 3N+1 C; vii) C gC = ; viii) A fB = ; Proof: In the proof Lemma 5.1.6, f',g' are constructed explicitly from f,g. Then A,B,C are used from Proposition

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2 C, and it is verified that A
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This note was uploaded on 08/05/2011 for the course MATH 366 taught by Professor Joshua during the Fall '08 term at Ohio State.

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6.3Refute071410 - 1 6.3 A Refutation In Proposition A can...

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