$2.$ Consider the neuron on the left with three inputs. The plot of the activation function of the neuron is given on the right. More precisely,

$0.$

v$<-0\mathrm{.}5$

$\}$

$+0\mathrm{.}5,$

$-0\mathrm{.}5$

v$0\mathrm{.}5$

f$($v$)=$

$1,$

$>0\mathrm{.}5$

$($a$)$ Suppose the error at the output is defined as E $=1($to$)$ where t is the target output. Derive the weight update equations for this neuron using the gradient descent method.

$($b$)$ Write a program to train this neuron such that it evaluates the function $(21$ AND $22)$ OR $23.$ Note that in this problem you are not restricted to use the McCulloch$-$Pitts neuron. In fact, you will train the network to find the weights. In your program, you should

i$.$ initialize the weights randomly,

ii$.$ use the weight update equations you derived in part $($a$).$

iii. stop the iterations when the error drops below the "machine epsilon" $($in MATLAB, it is denoted by the constant eps.$).$

Report the resulting values of the weights. How does the learning rate affect the convergence? Try different values and discuss.

$($c$)$ Since there are three inputs, they can be visualized in $3$D space. Plot the input points and the discriminant corresponding to the weights you found in part $($b$),$ and explain how this problem can be interpreted as a linear discriminant problem separating two sets of points.