Final answer:
The VC dimension of the triangle hypothesis class is 7, shown by placing 7 equidistant points on a circle and demonstrating that all labeling combinations can be separated by triangles, while no set of 8 points can achieve this.
Step-by-step explanation:
To show that the VC dimension of the triangle hypothesis class is 7 in two dimensions, one must demonstrate that there exists a set of 7 points which can be shattered by the hypothesis class, and no set of 8 or more points can achieve this. Shattering means that for every possible combination or labeling of the points into two classes, there exists a triangle that can separate the points according to that combination.
An effective way to prove the VC dimension of 7 is to place the seven points equidistant on a circle. This configuration guarantees that no three points are collinear, and it also allows the combination of any three points to form a triangle that can separate the classes in any desired labeling.
By drawing triangles to encompass different combinations of these labeled points, you can indeed create classifications for all possible arrangements of the 7 points. Determining that no set of 8 points can be shattered this way, since there would always be an arrangement of labeling that cannot be separated by a single triangle, confirms that the VC dimension of the triangle hypothesis class in two dimensions is 7.