Consider the process of building a decision$-$tree based classifier using Entropy as a measure of impurity associated with a tree node that represents a subset of training examples. A node is split into

partitions represented by its child nodes based on the values of a selected attribute. The goodness of the attribute for the split, referred to as gain of the attribute, is estimated in terms of the difference

between the impurity of the parent node and the weighted sum of the impurities of the child nodes.

Consider the training set of examples described in terms of three attributes $A_1,A_2$ and $A_3$ in addition to the distribution of examples among four classes namely $C_1,C_2,C_3$ and $C_4.$

Each row of the table describes a subset of examples in terms of the three binary attributes $A_1,A_2$ and $A_3$ and indicates the number of examples among the subset that fall into class $C_1,C_2,C_3$ and $C_4$

in terms of the values given in the last four columns. The classifier aims at classifying the training examples into one among the predefined set of classes.

Estimate the goodness of the attributes for splitting the training set given in the table. Which of the following is TRUE about the goodness of attributes for a split.

a$.$ Splitting based on attribute $A_2$ is the better way to partition the training set.

b$.$ The value of the information gain of attribute $A_2$ is greater than $0\mathrm{.}5$ and less than $0\mathrm{.}6.$

c$.$ The value of the gain ratio of attribute $A_2$ is greater than $0\mathrm{.}2.$

d$.$ The value of the information gain of attribute $A_1$ is greater than $0\mathrm{.}5$ and less than $0\mathrm{.}7.$