Question $3$: Neural Networks

Consider the Binary function $y=f_3(x_1,x_2,x_3),$ where $x_{i}in\{0,1\}.$

$($a$)$ Design a single$-$unit Perceptron that will classify a function of the type repre$-$

sented by $f_3.$ Draw a diagram to show the structure, parameters and function$($s$)$

of this Perceptron. clearly showing how the output value is arrived at$.$

$($b$)$ The NAND Boolean function is defined as NOT$(AND).$ Write down the

truth table for the function $f_4(x_1,x_2,x_3)=$NAND$(x_1,x_2,x_3).$

$($c$)$ Calculate the output values generated by the Perceptron for all possible values

of $x_i$ in $f_3(),$ using the ReLu activation function and the following weight values:

$w_1=-1\mathrm{.}0$

$w_2=-1\mathrm{.}0$

$w_3=-1\mathrm{.}0$

$b=3\mathrm{.}0$

Show ALL your calculations.

$($d$)$ What effects do changes in the bias value of a neuron in a feedforward neural

network have on the position of the decision boundary$/$hyperplane$?$ Be specific

about the case where bias $=0.$