Definition
Suppose we have a class C of 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
Equivalently, C shatters A when the power set P(A) is the set { U ∩ A | U ∈ C }.
For example, the class C of all discs in the plane (two-dimensional space) cannot shatter every set A of four points, yet the class of all convex sets in the plane shatters every finite set on the (unit) circle. (For the collection of all convex sets, connect the dots!)
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 because, in empirical processes, we are interested in the shattering of finite sets of data points.
Read more about this topic: Shattered Set
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