Consider the $8$ instances below, with positive instances marked as Pi and negative

instances as Ni

:

P$1=(-3,-1)$

P$2=(-1,2)$

P$3=(1,3)$

P$4=(3,1)$

N$1=(-2,-3)$

N$2=(0\mathrm{.}5,-3\mathrm{.}5)$

N$3=(4,-2)$

N$4=(2,-1)$

Figure $1$ shows the instances as well as the decision surface$/$hyperplane for a

single neuron Perceptron.

$5432112345$

$5$

$4$

$3$

$2$

$1$

$1$

$2$

$3$

$4$

$5$

$0$

N$4$

N$3$

N$1$

N$2$

P$4$

P$3$

P$2$

P$1$

X

Y

Figure $1$: $8$ instances and the hyperplane of a Perceptron.

The cut$-$off points of the hyperplane on the axes are at $(2,0)$ and $(0,-1).$

$($a$)$ Calculate the values of the weights of the Perceptron, using the position of the

decision surface$/$hyperplane$.$

$($b$)$ Design a neural network, using one or more Perceptron neurons, for the

Boolean function f$1,$ as defined in Table $5.$

x y f$1($x$,$ y$)$

$-2-2-1$

$-2+2+1$

$+2-2+1$

$+2+2+1$

Table $5$: Function f$1($x$,$ y$).$

Use the threshold activation function $($as defined in the Appendix on p$.7)$ in

your calculations. Plot the points of Table $5$ in the input space of f$1.$ Show the

decision boundary used in your calculations. Calculate the weight values for

the connections in your neural network.