Editing Shattered set (section)

==Example==

We will show that the class of all [[disc (geometry)|discs]] in the [[plane (geometry)|plane]] (two-dimensional space) does not shatter every set of four points on the [[unit circle]], yet the class of all [[convex set]]s in the plane does shatter every finite set of points on the [[unit circle]].

Let ''A'' be a set of four points on the unit circle and let ''C'' be the class 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 where ''C'' shatters ''A'', 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. Any attempt to include those points on the opposite side will necessarily include other points not in that pair.  Hence, any pair of opposite points cannot be isolated out of ''A'' using intersections with class ''C'' and so ''C'' does not shatter ''A''. 

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 that were formerly problematic. 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).