$($a$)$ Expand the following binary decision tree. $(3)$

A

$0$ B

$01$

$01$

$01$

Figure $1$: Decision tree for f$1($A$,$ B$).$

$($b$)$ Derive a Boolean function for the binary decision tree in Figure $1.$ Use a truth

table and a Karnaugh map to derive your answer. $(7)$

$($c$)$ Convert the decision tree in $($a$)$ to a decision list. $(3)$

$($d$)$ Calculate Entropy$($V $)$ for the data in the Table $1.$ Show your assumptions and

define your notation clearly $($use Table $4$ on p$.6$ in your calculations$).(5)$

I S C F V

i$1$ m b b T

i$2$ s r s T

i$3$ l g p T

i$4$ l g s T

i$5$ s r w F

i$6$ l r w F

i$7$ l r p F

Table $1$: Objects with attributes and their classification as V $($T$,$ F$).$

$($e$)$ What is the purpose of using Information Gain in the construction of decision

trees? $(2)$

$($f$)$ Calculate the Information Gain for the F attribute in Table $1.(8)$

$($g$)$ Given the following Information Gain values:

InformationGain$($V$,$ C $)=0\mathrm{.}799$

InformationGain$($V$,$ F$)=0\mathrm{.}286$

which attribute becomes the root node for a decision tree for the data in

Table $1?$ Explain why this attribute is the correct root node.