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The concept of '''shattered sets''' plays an important role in [[Vapnik&ndash;Chervonenkis theory]], also known as VC-theory. Shattering and VC-theory are used in the study of [[empirical process]]es as well as in statistical [[computational learning theory]].

==Definition==
Suppose we have a [[class (set theory)|class]] ''C'' of [[Set (mathematics)|sets]] and a given set ''A''. ''C'' is said to  ''shatter''  ''A'' if, for each subset ''T'' of ''A'', there is some element ''U'' of ''C'' such that 

: <math> U \cap A = T.\,</math>

Equivalently, ''C'' shatters ''A'' when the [[power set]] ''P''(''A'') is the set { ''U'' ∩ ''A'' | ''U'' ∈ ''C'' }.

For example, the class  ''C''  of all [[disc (geometry)|discs]] in the [[plane (geometry)|plane]] (two-dimensional space) cannot shatter every set ''A'' of four points, yet the class of all [[convex set]]s in the plane shatters every finite set on the (unit) circle. (For the collection of all convex sets, connect the dots!<!-- Is this metaphorical or literary?--><!-- Neither. It's a proof of the second assertion.-->)

We employ the letter  ''C''  to refer to a "class" or "collection" of sets, as in a Vapnik&ndash;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.

==Example==
Suppose we have a set ''A'' of four points on the unit circle, and we wish to know if it is shattered by the class ''C'' of all discs.
[[File:shattering01.png|thumb|right|300px|The set ''A'' of four particular points on the unit circle. (the unit circle is shown in purple))]]
To test this, we attempt to draw a disc around every subset of points in ''A''. First, we draw a disc around the subsets of each isolated point. Next, we try to draw a disc around every subset of point pairs. This turns out to be doable for adjacent points, but impossible for points on opposite sides of the circle. As visualized below:

<gallery>
Image:shattering02.png|Each individual point can be isolated with a disc (showing all four).
Image:shattering03.png|Each subset of adjacent points can be isolated with a disc (showing one of four).
Image:shattering04.png|A subset of points on opposite sides of the unit circle can ''not'' be isolated with a disc.
</gallery>

Because there is some subset which can ''not'' be isolated by any disc in ''C'', we conclude then that ''A'' is not shattered by ''C''. And, with a bit of thought, we can prove that NO set of four points is shattered by this ''C''.

However, if we redefine ''C'' to be the class of all ''elliptical discs'', we find that we can still isolate all the subsets from above, as well as the points which were problems. Thus, this specific set of 4 points is shattered by the class of elliptical discs. Visualized below:

<gallery>
Image:shattering05.png|Opposite points of ''A'' are now separable by some ellipse (showing one of two)
Image:shattering06.png|Each subset of three points in ''A'' is also separable by some ellipse (showing one of four)
</gallery>

With a bit of thought, we could generalize that any set of finite points on a unit circle could be shattered by the class of all [[convex set]]s (visualize connecting the dots).

==Shatter coefficient==
To quantify the richness of a collection ''C'' of sets, we use the concept of ''shattering coefficients'' (also known as ''shatter coefficients'' or the ''growth function''). For a collection ''C'' of sets s⊂Ω,  Ω being any space,  often a [[probability space]], we define
the ''n''<sup>th</sup> ''shattering coefficient'' of ''C'' as

:<math> S _C(n) = \max_{x_1,x_2,\dots,x_n \in \Omega } \operatorname{card} \{\,\{\,x_1,x_2,\dots,x_n\}\cap s, s\in  C \}</math>

where <math>\operatorname{card}</math> denotes the [[cardinality]] of the set.

<math> S _C(n) </math>  is equal to the largest number of subsets of any set ''A''  of ''n'' points that can be formed by intersecting  ''A'' with the sets in collection ''C''.

Here are some facts about <math>S_C(n)</math>:
:1.<math>S_C(n)\leq 2^n</math> for all ''n'' because <math>\{s\cap A|s\in C\}\subseteq P(A)</math> for any <math>A\subseteq \Omega</math>.
:2. If <math>S_C(n)=2^n</math>, that means there is a set of cardinality ''n'', which can be shattered by ''C''.
:3. If <math>S_C(N)<2^N</math> for some <math>N>1</math> then <math>S_C(n)<2^n</math> for all <math>n\geq N</math>.
The third property means that if ''C'' cannot shatter any set of cardinality ''N'' then it can not shatter sets of larger cardinalities.

==Vapnik&ndash;Chervonenkis class==
The [[VC dimension]] of a class ''C'' is defined 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>

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&ndash;Chervonenkis class'' or ''VC class''. A class ''C'' is [[Glivenko&ndash;Cantelli class|uniformly Glivenko&ndash;Cantelli]] if and only if it is a VC class.

==See also==
*[[Sauer–Shelah lemma]], relating the cardinality of a [[family of sets]] to the size of its largest shattered set

==References==
*{{citation|last1=Wencour|first1=R. S.|first2=R. M.|last2=Dudley|title=Some special Vapnik–Chervonenkis classes|journal=Discrete Mathematics|volume=33|year=1981|pages=313–318|issue=3|doi=10.1016/0012-365X(81)90274-0}}.
*{{citation|authorlink=J. Michael Steele|last=Steele|first=J. M.|year=1975|title=Combinatorial Entropy and Uniform Limit Laws|series=Ph.D. thesis|publisher=Stanford University, Mathematics Department}}
*{{citation|authorlink=J. Michael Steele|last=Steele|first=J. M.|year=1978|title=Empirical discrepancies and subadditive processes|journal=Annals of Probability|volume=6|pages=118–227|issue=1|jstor=2242865}}.

==External links==
*[http://www-stat.wharton.upenn.edu/~steele/Rants/ShatteredSets.html Origin of "Shattered sets" terminology], by J. Steele


[[Category:Empirical process]]
[[Category:Computational learning theory]]