A shattered set relative to a collection 
of subsets of a set 
is a subset 
such that each subset 
can be expressed as the intersection of 
with a subset 
in 
.
Symbolically, then, 
is shattered by 
if for all 
,
there exists some 
for which 
.
If 
is shattered by 
, one says that 
shatters 
.
There are a number of equivalent ways to define shattering. One can easily verify that the above definition is equivalent to saying that 
shatters 
if
 |
where 
denotes the power set of 
. Yet another way to express this concept is to say that a
set 
of cardinality 
is shattered by a set 
if 
where here,
 |
In the field of machine learning theory, one usually considers the set 
to be a sample of outcomes
drawn according to a distribution 
with the set 
representing a collection of
"known" concepts or laws. In this context,
saying that 
is shattered by 
intuitively means that all of the constituent outcomes
in 
can be known by knowing only the laws
in 
.
