Consider a training set sampled uniformly from the two$-$dimensional space shown in Figure $1.$ Assume that the training set size is large enough so that the probabilities can be calculated accurately based on the areas of the selected regions. The space is divided into three classes A$,$ B$,$ and C$.$

$($a$)$ Compute the entropy for the overall data.

$($b$)$ Compare the entropy when the data is split at x x and y

$($c$)$ Based on your answer in part $($b$),$ which attribute split condition

should be used as the root of the decision tree.

$($d$)$ Draw the full decision tree for the data set. Constraints: Use only the decision boundaries x $=0\mathrm{.}7,$ x $=0\mathrm{.}2,$ y $=0\mathrm{.}6,$ y $=0\mathrm{.}3,$ y $=0\mathrm{.}8$ to contruct your tree. Each level of the tree will maximize the information gain.

Figure $1$: $2$D Region