You are given a three$-$dimensional binary classification data set with six data points.

You use the three binary features $,$ and $\{$:$x_3)$ to predict the class label using a

decision tree classifier. You use the entropy as the impurity measure. In the case of

ties when splitting, prefer using the feature with the smaller index $(x_1$ over $x_2,x_1$ over

$x_3,$ and $x_2$ over $x_3).$ You might find the following values useful.

$lo{g}_2(1)=0\mathrm{.}00,lo{g}_2(2)=1\mathrm{.}00,lo{g}_2(3)=1\mathrm{.}58,lo{g}_2(4)=2\mathrm{.}00$

$lo{g}_2(5)=2\mathrm{.}32,lo{g}_2(6)=2\mathrm{.}58,lo{g}_2(7)=2\mathrm{.}81,lo{g}_2(8)=3\mathrm{.}00$

$($a$)$ Which variable would you use in the first split at the root node? Justify your

answer.

$($b$)$ Construct the whole decision tree learned on this data set.