By Laurence Sigler
First released in 1202, Fibonacci’s Liber Abaci used to be probably the most vital books on arithmetic within the heart a while, introducing Arabic numerals and strategies all through Europe. this is often the 1st translation right into a smooth ecu language, of curiosity not just to historians of technology but additionally to all mathematicians and arithmetic academics attracted to the origins in their tools.
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Additional info for Fibonacci's Liber Abaci
Any sigma-field ℱ is closed under countable intersections because ∩i=1∞Ei=(∪i=1∞EiC)C and each EiC belongs to ℱ. Because any field or sigma-field ℱ is nonempty, it contains a set E, and therefore EC. Consequently, any field or sigma-field contains Ω=E∪EC and ∅=ΩC. A sigma-field is more restrictive than a field. Because we want to be able to consider countable unions, we require the allowable sets ℱ to be a sigma-field. Therefore, it is understood in probability theory and throughout the remainder of this book that we are allowed to consider only the events belonging to the sigma-field ℱ.
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It turns out that the tersest conditions yielding the sets we would like to be able to consider are given in the following definition. 2. Field and sigma-field A nonempty collection of subsets of Ω is called a: Field (also called an algebra) if ℱ is closed under complements and unions of pairs of members. That is, whenever E∈ℱ, then EC∈ℱ, and whenever E1,E2∈ℱ, then E1∪E2∈ℱ. Sigma-field (also called a sigma-algebra) if ℱ is closed under complements and countable unions. ∈ℱ, then ∪i=1∞Ei∈ℱ. 3. Any field is also closed under finite unions because we can apply the paired union result repeatedly.