9. 十九世纪下半叶集合论的创立奠定了现代数学的基础.著名的“康托三分集”是数学理性思维的构造产物,具有典型的分形特征.仿照“康托三分集”我们可以构造一个“四分集”,其操作过程如下:将闭区间
![](http://math.21cnjy.com/mml2svg?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmrow%3E%3Cmo+stretchy%3D%22false%22%3E%5B%3C%2Fmo%3E%3Cmn%3E0%3C%2Fmn%3E%3Cmo%3E%2C%3C%2Fmo%3E%3Cmn%3E1%3C%2Fmn%3E%3Cmo+stretchy%3D%22false%22%3E%5D%3C%2Fmo%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
均分为四段,去掉其中的区间段
![](http://math.21cnjy.com/mml2svg?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmrow%3E%3Cmrow%3E%3Cmo%3E%28%3C%2Fmo%3E%3Cmrow%3E%3Cmfrac%3E%3Cmn%3E1%3C%2Fmn%3E%3Cmn%3E4%3C%2Fmn%3E%3C%2Fmfrac%3E%3Cmo%3E%2C%3C%2Fmo%3E%3Cmfrac%3E%3Cmn%3E1%3C%2Fmn%3E%3Cmn%3E2%3C%2Fmn%3E%3C%2Fmfrac%3E%3C%2Fmrow%3E%3Cmo%3E%5D%3C%2Fmo%3E%3C%2Fmrow%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
记为第一次操作;再将剩下的三个区间
![](http://math.21cnjy.com/mml2svg?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmrow%3E%3Cmrow%3E%3Cmo%3E%5B%3C%2Fmo%3E%3Cmrow%3E%3Cmn%3E0%3C%2Fmn%3E%3Cmo%3E%2C%3C%2Fmo%3E%3Cmfrac%3E%3Cmn%3E1%3C%2Fmn%3E%3Cmn%3E4%3C%2Fmn%3E%3C%2Fmfrac%3E%3C%2Fmrow%3E%3Cmo%3E%5D%3C%2Fmo%3E%3C%2Fmrow%3E%3Cmo%3E%2C%3C%2Fmo%3E%3Cmrow%3E%3Cmo%3E%28%3C%2Fmo%3E%3Cmrow%3E%3Cmfrac%3E%3Cmn%3E1%3C%2Fmn%3E%3Cmn%3E2%3C%2Fmn%3E%3C%2Fmfrac%3E%3Cmo%3E%2C%3C%2Fmo%3E%3Cmfrac%3E%3Cmn%3E3%3C%2Fmn%3E%3Cmn%3E4%3C%2Fmn%3E%3C%2Fmfrac%3E%3C%2Fmrow%3E%3Cmo%3E%5D%3C%2Fmo%3E%3C%2Fmrow%3E%3Cmo%3E%2C%3C%2Fmo%3E%3Cmrow%3E%3Cmo%3E%28%3C%2Fmo%3E%3Cmrow%3E%3Cmfrac%3E%3Cmn%3E3%3C%2Fmn%3E%3Cmn%3E4%3C%2Fmn%3E%3C%2Fmfrac%3E%3Cmo%3E%2C%3C%2Fmo%3E%3Cmn%3E1%3C%2Fmn%3E%3C%2Fmrow%3E%3Cmo%3E%5D%3C%2Fmo%3E%3C%2Fmrow%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
,分别均分为四段,并各自去掉第二个区间段,记为第二次操作;···如此这样,每次在上一次操作的基础上,将剩下的各个区间分别均分为四段,同样各自去掉第二个区间段.操作过程不断地进行下去,以至无穷,剩下的区间集合即是“四分集”.若使去掉的各区间长度之和不小于
![](http://math.21cnjy.com/mml2svg?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmrow%3E%3Cmfrac%3E%3Cmrow%3E%3Cmn%3E19%3C%2Fmn%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmn%3E20%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmfrac%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
,则需要操作的次数
n的最小值为(参考数据:
![](http://math.21cnjy.com/mml2svg?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmrow%3E%3Cmi%3Elg%3C%2Fmi%3E%3Cmn%3E2%3C%2Fmn%3E%3Cmo%3E%E2%89%88%3C%2Fmo%3E%3Cmn%3E0.3010%3C%2Fmn%3E%3Cmo%3E%2C%3C%2Fmo%3E%3Cmi%3Elg%3C%2Fmi%3E%3Cmn%3E3%3C%2Fmn%3E%3Cmo%3E%E2%89%88%3C%2Fmo%3E%3Cmn%3E0.4771%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
)( )