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Shattered set
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==Definition== Suppose ''A'' is a [[set (mathematics)|set]] and ''C'' is a [[class (set theory)|class]] of sets. The class ''C'' '''shatters''' the set ''A'' if for each subset ''a'' of ''A'', there is some element ''c'' of ''C'' such that : <math>a = c \cap A.</math> Equivalently, ''C'' shatters ''A'' when their [[Growth function#Definitions|intersection]] is equal to ''A'''s [[power set]]: ''P''(''A'') = { ''c'' β© ''A'' | ''c'' β ''C'' }. We employ the letter ''C'' to refer to a "class" or "collection" of sets, as in a Vapnik–Chervonenkis class (VC-class). The set ''A'' is often assumed to be [[finite set|finite]] because, in empirical processes, we are interested in the shattering of finite sets of data points.
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