How to prove that the XOR problem for dimension d is not lineary seperable

Question

The 2 dimensions xor problem can be converted to 4 equations which is possible to prove that are not possible to solve

  x1       x2     output
  0         0         0
  0         1         1
  1         0         1
  1         1         0

w1*0 + w2*0 <= 0
w1*0 + w2*1 > 0
w1*1 + w2*0 > 0
w1*1 + w2*1 <= 0

How can I prove that the XOR problem for dimension d is not lineary seperable? How to relate to an even d and an odd d?

I thought of the following answer: Lets observe all of the equations of the form:

0*w1 + .. + 1*wi + .. + 0*wd > 0, for each i=1..d

These equations obligate wi > 0, for each i.

Now lets take the last equation.

1*w1 + 1*w2 + ... + 1*wd <= 0 (only when d is even)

This equation force wi<=0 for all i.

So:

wi > 0, for each i.
wi<=0 for all i.

Cannot be solved because. On the odd - d case, we'll have to consider the all of the equations with 1 zero (d equations), and it will get to the same contradiction.

But - i'm not sure it's the good-practice way.

Thanks

Answer 1

Your answer for 2 dimensions can be true even for $d>2$.

Consider the vectors:
$(1, 0, 0, ..., 0)$
$(0, 1, 0, ..., 0)$
$(1, 1, 0, ..., 0)$

You can't have $w1, w2$ s.t.
$w_1>0, w_2 >0$
$w_1 + w_2 \leq 0$

edit:
this is for the equations that you posted for when $b=0$,
for $b \neq 0$, see Arthur's answer

Answer 2

If $XOR_d(x_1,x_2,\ldots,x_d)$ is the (linear) XOR function on $d$ variables (defined as 1 iff an odd number of the inputs are 1), then define the two functions $$ f(x_1)=XOR_d(x_1,0,0,\ldots,0)=w_1x_1+a\\ g(x_1)=XOR_d(x_1,0,0,\ldots,0,1)=w_1x_1+b $$ But $f$ is (strictly) increasing and $g$ is (strictly) decreasing, which is impossible.