Spivak's Calculus Prologue Problem 3 (v)

samedi 2 novembre 2013

Here's my attempt at this proof. Is this correct?





1. The problem statement, all variables and given/known data



Prove [itex]\frac{a}{b}[/itex][itex]/[/itex][itex]\frac{c}{d}[/itex] = [itex]\frac{ad}{bc}[/itex]



2. Relevant equations



P 1-12



3. The attempt at a solution



[itex]\frac{a}{b}[/itex][itex]/[/itex][itex]\frac{c}{d}[/itex] = [itex]\frac{ad}{bc}[/itex]



[itex]\frac{a}{b}[/itex][itex]/[/itex][itex]\frac{c}{d}[/itex] = (ad)(bc)[itex]^{-1}[/itex]



[itex]\frac{a}{b}[/itex][itex]/[/itex][itex]\frac{c}{d}[/itex] = (ad)(b[itex]^{-1}[/itex]c[itex]^{-1}[/itex])



[itex]\frac{a}{b}[/itex][itex]/[/itex][itex]\frac{c}{d}[/itex] = (ab[itex]^{-1}[/itex])(dc[itex]^{-1}[/itex])



[itex]\frac{a}{b}[/itex][itex]/[/itex][itex]\frac{c}{d}[/itex] = (ab[itex]^{-1}[/itex])(d[itex]^{-1}[/itex]c)[itex]^{-1}[/itex]



[itex]\frac{a}{b}[/itex][itex]/[/itex][itex]\frac{c}{d}[/itex] = [itex]\frac{a}{b}[/itex][itex]/[/itex][itex]\frac{c}{d}[/itex]






Also, do proofs have to be in if, then, hence form like they are when Spivak is presenting the basic properties of numbers?






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