Question $2$

Design neural networks, using one or more Perceptron neurons, for the data as detailed below.

Let your Perceptron use the threshold activation function:

$f_{\text{t}\text{h}\text{r}\text{r}\text{s}\text{s}\text{h}\text{o}\text{l}\text{d}}(z)=\{\begin{array}{cc}1,& z0\\ 0,& z<mn\; id="EditorElement\_714812">0</mn>\end{array}$

Figure $4$: Threshold activation function.

For each sub$-$question, draw diagrams to show the decision boundaries that your network uses, and

the structure of the neural network showing the weights that will correctly classify the function. Show

the arguments for your choices, all your assumptions, definitions, and calculations. Prove that the

weights you use will correctly classify the function.

Note that this question does not ask you to train a neural network using one of the neural network

algorithms. The purpose here is similar to the question on the relationship between decision trees

and Boolean expressions, in the previous assignment. It is possible to directly map a specific

Boolean expression to a network of Perceptrons. Rojas has a very good explanation of the concepts

needed in Chapter $2($and some aspects on XOR in Chapter $6).$

Question $2($a$)$

Data set $1$:

$P_1=(-3,3)$

$P_2=(3,-3)$

$P_3=(3,3)$

$N_1=(-3,-3)$ Question $2($b$)$

Data set $2$:

$P_1=(-3,-3)$

$P_2=(3,3)$

$N_1=(-3,3)$

$N_2=(3,-3)$

Mark out of $100.$

$40$ or less for clear indication that student does not understand the topic or evidence of plagiarism

$50$ for a fair understanding

$60-70$ for understanding and clear well defined examples

$80+$ for exceptional detail