$($a$)$ Convert the following decision list into a binary decision tree. $(4)$

$$ if A $=1$ then $1$

$$ else if B $=0$ then $1$

$$ else $1$

$($b$)$ Write down the truth table for the decision list in $($a$),$ and derive a Boolean

function from the truth table. $(5)$

$($c$)$ What is meant by the term entropy? Define the entropy of a data set S

in mathematical terms. Why is Entropy used as a measure in constructing

Decision Trees? $(5)$

$($d$)$ Calculate Entropy$($C ancer$)$ for the data in the Table $1.$ Show your assumptions and define your notation clearly $($use the Table on p$.7$ in your calculations$).$

Show all your calculations. $(5)$

sample Size Type Other C ancer

s$1$ M macule flat Y

s$2$ S tumour sharp Y

s$3$ L plaque elevated Y

s$4$ L plaque sharp Y

s$5$ S tumour localised N

s$6$ L tumour localised N

s$7$ L tumour elevated N

Table $1$: Objects with attributes and their classification as C ancer$($Y$,$ N$).$

$($e$)$ Calculate the Information Gain for the Type attribute in Table $1.(6)$

$($f$)$ Without doing any further calculations, which attribute would be used for the

root node of the decision tree?F$15$