F Dimension is introduced by Vladimir Vapnik and Alexi Shenis in 1970, and it has become one of the fundamental concepts in statistical learning theory. F dimension of a hypothesis set \( H \) is the largest number of data points that can be shattered by \( H \). Now, the question is: What is "shattered" in the context of a hypothesis set \( H \)? A hypothesis set \( H \) shatters a set of points \( S \) if, for all possible labelings of the data points, there exists a hypothesis in \( H \) that correctly classifies the data points. Let's assume that we have two data points in this case. The two data points can have different labelings. For example, both the data points may be positive, one is positive, one is negative, or both may be negative. If there is a hypothesis in the hypothesis space \( H \) that classifies all possible labelings correctly, then you can say that the classifier shatters the given data points. The largest number of data points which were classified correctly is nothing but the VC dimension in this case. now we will take some example to understand the VC Dimension initially I will consider the straight line as a classifier here so before that we will try to understand the number of labelings or the classifiers here if you have n number of data points then there exist 2 to n different classifiers or the different labelings in this case now if you have two data points there exist four different classifiers or the labels you can notice here I have considered the two notations that is green and red here green is nothing but positive red is nothing but a negative class here in the first case both of them are positive in the second case one is positive and one is negative third case one is positive and one is negative in the fourth case you can see here both of them are negative in this case now because we are considering straight line as a classifier Here, in this case, we can draw a straight line, something like this. The meaning of this is that above this straight line, all the examples belong to the positive class, and below this line, all of them belong to the negative class. In this case, we are able to classify them correctly. Similarly, for the remaining possibilities, we can also draw a straight line so that we can classify the examples correctly. This shows that if you have two data points, with the help of a straight line, we can classify them correctly in this case. Now, coming back to the next possibility, let's assume that we have three data points. There are 8 different classifiers or labelings that exist here, as shown in this diagram. These labelings are: all three are positive, two are positive and one is negative, two are positive and one is negative, and lastly, all of them are negative. In this case, we are considering the straight line as a classifier. For the first labeling, if you draw the straight line something like this, above this straight line, all of them are positive examples, and below this, all of them are negative examples. We can draw a straight line so that we can classify the examples correctly here. The same thing can be done for all of these particular possibilities. The meaning of this one is we have a hypothesis in hypothesis space, which classifies the three data points correctly with the help of a straight line in this case. Coming back to the next one, that is four data points. If you have four data points, the total number of possibilities are 16 here, as shown in this diagram. In this case, some of the labelings were classified correctly, while some of them cannot be classified. For example, we can classify these labelings correctly here, and we can classify these labelings also. But if you look at this one, we cannot draw a straight line to classify these examples correctly. We cannot draw it here, we cannot draw it here, or we cannot draw something like this. If you draw in any way, some of the examples will be incorrectly classified in this case.