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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

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enter image description here then:

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if there are no contradictions in the previous inequalities, the problem is linearly separable, as observed from the inequalities 2, 3, 4 is impossible

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and that its sum is less than zero

my question is how that result of contradiction(inequalities 2,3,4)

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1 Answer 1

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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 $b<0$) $$ W_{1,1}+W_{2,1}+b \geq W_{1,1}+W_{2,1}+2b \geq 0 $$

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  • $\begingroup$ then the contradiction is W1,1+W2,1+b<=0 and W1,1+W2,1+2b≥0 $\endgroup$
    – userStack
    Apr 15, 2018 at 19:41
  • $\begingroup$ please you could explain a little more the last part $\endgroup$
    – userStack
    Apr 15, 2018 at 19:49
  • $\begingroup$ @x-rw At first ($p_1$) says $b$ is negative. So $b\geq 2b$. OK? Now add ($p_2$) and ($p_3$), then you have $W_{1,1}+W_{2,1}+2b\geq0$. However $b\geq 2b$ so that $W_{1,1}+W_{2,1}+b\geq 0$ too. But it contradict to ($p_4$). $\endgroup$
    – ChoF
    Apr 15, 2018 at 22:22
  • $\begingroup$ how to come to the conclusion of b>=2b $\endgroup$
    – userStack
    Apr 16, 2018 at 1:06
  • $\begingroup$ @x-rw From ($p_1$) we know $b$ is negative. So $b\geq 2b$. For example, $-1>-2$. $\endgroup$
    – ChoF
    Apr 16, 2018 at 1:10

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