To compare the capabilities of some popular classification algorithms, provide sample datasets that cannot be dealt with accurately by one algorithm but can be accurately dealt with by the other. In every case, explain why one of the algorithms fails to discover the correct clusters or classes.

a$)$ Write a $2-$dimensional dataset of ten points with two classes that can be classified with $100\%$ accuracy by a decision tree, but not by a logistic regression classifier. $($Please describe every data point and its class label numerically. A drawing of data points without numerical values of attributes will not get any mark.$)(5$ marks$)$

Draw the data points in a$),$ and also draw the decision boundaries of a decision tree and a logistic regression classifier to explain why the dataset in a$)$ can be $100\%$ accurately classified by a decision tree but cannot be $100\%$ correctly classified by a logistic regression classifier. $($The drawing is only required to be qualitatively correct$)(5$ marks$)$

c$)$ Write a $2-$dimensional dataset of ten points with two classes that can be classified with $100\%$ accuracy by a logistic regression classifier, but not by a $3-$Nearest Neighbors classifier. $($Please describe every data point and its class label numerically. A drawing of data points without numerical values of attributes will not get any mark. $(5$ marks$)$

d$)$ Draw the data points in c$),$ and also draw the decision boundaries of a logistic regression classifier and a $3-$Nearest Neighbors classifier to explain why the dataset in c$)$ can be $100\%$ accurately classified by a logistic regression classifier but cannot be $100\%$ correctly classified by a $3-$Nearest Neighbors classifier. $($The approximate decision boundaries are only required to be qualitatively correct$)(5$ marks$)$