I was reading a book of neural networks in Spanish and I do not understand the inequalities on page 26
the book: http://disi.unal.edu.co/~lctorress/RedNeu/LiRna004.pdf
I have a table of XOR and inequality restrictions
then:
if there are no contradictions in the previous inequalities, the problem is linearly separable, as observed from the inequalities 2, 3, 4 is impossible
and that its sum is less than zero
my question is how that result of contradiction(inequalities 2,3,4)
Let rewrite the inequalities: $$ \begin{align*} 0 * W_{1,1} + 0 * W_{2,1} + b &< 0 \tag{$p_1$} \\ 0 * W_{1,1} + 1 * W_{2,1} + b &\geq 0 \tag{$p_2$} \\ 1 * W_{1,1} + 0 * W_{2,1} + b &\geq 0 \tag{$p_3$}\\ 1 * W_{1,1} + 1 * W_{2,1} + b &< 0 \tag{$p_4$} \\ \end{align*} $$
From ($p_1$), we have $b<0$. From ($p_2$) and ($p_3$), we have $W_{1,1}+b\geq 0$ and $W_{2,1}+b\geq0$ so that $W_{1,1}+W_{2,1}+2b\geq 0$. However, ($p_4$) implies a contradiction $W_{1,1}+W_{2,1}+b<0$, because (as $b<0$) $$ W_{1,1}+W_{2,1}+b \geq W_{1,1}+W_{2,1}+2b \geq 0 $$