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Shattered set
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==Vapnik–Chervonenkis class== {{Main|VC dimension}} If ''A'' cannot be shattered by ''C'', there will be a smallest value of ''n'' that makes the shatter coefficient(n) less than <math>2^n</math> because as ''n'' gets larger, there are more sets that could be missed. Alternatively, there is also a largest value of ''n'' for which the S_C(n) is still <math>2^n</math>, because as ''n'' gets smaller, there are fewer sets that could be omitted. The extreme of this is S_C(0) (the shattering coefficient of the empty set), which must always be <math>2^0=1</math>. These statements lends themselves to defining the [[VC dimension]] of a class ''C'' as: :<math>VC(C)=\underset{n}{\min}\{n:S_C(n)<2^n\}\,</math> or, alternatively, as :<math>VC_0(C)=\underset{n}{\max}\{n:S_C(n)=2^n\}.\,</math> Note that <math>VC(C)=VC_0(C)+1.</math>. The VC dimension is usually defined as VC_0, the largest cardinality of points chosen that will still shatter ''A'' (i.e. ''n'' such that <math>S_C(n)=2^n</math>). Altneratively, if for any ''n'' there is a set of cardinality ''n'' which can be shattered by ''C'', then <math>S_C(n)=2^n</math> for all ''n'' and the VC dimension of this class ''C'' is infinite. A class with finite VC dimension is called a ''Vapnik–Chervonenkis class'' or ''VC class''. A class ''C'' is [[Glivenko–Cantelli class|uniformly Glivenko–Cantelli]] if and only if it is a VC class.
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