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Vito Prosciutto: Teaching community college math on the road to a PhD.

Monday, May 02, 2005

21:45
Proving that 1 is positive 

I'm getting a late start on my plan to do the homework for the grad courses that I'm (not) taking this semester. And early on, I hit a snag. Exercise 1 in chapter 2: Prove that 1 in P (P being the set of positive numbers). It looks to me like Royden's axioms for the positive numbers are incomplete: The notation is a bit sloppy already with B4 (I would write "either" after the implication). But it seems to be the case that these axioms also apply for P being the set of negative real numbers. I'll have to dig up the argument for (-1)(-1)=1 from the Art of the Infinite and see if this helps. It's always these trivial problems at the beginning of the book which trip me up... Update (5/3): Aha, I worked it through and it seems that part of the proof that 1 in P required me to show that the product of two negative numbers is a positive number. That then allowed me to use the contrapositive of B2 to show that if xy not in P that exactly one of x and y is not in P. From there, I could take the fact that given x in P, 1(-x)=-x requires 1 in P. I guess it's questions like this that separate the math majors from the English majors.

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