Perceptrons: Functions not learnable without bias I am trying to determine which functions are not learnable without a bias when building a perceptron. The set of functions I need to evaluate is {NOT, OR, AND}. 

Could someone help interpret these results? 

AND FUNCTION

https://imgur.com/a/OTFUT

OR FUNCTION

I get a divide by 0 error in my script.

NOT FUNCTION

https://imgur.com/a/wZnxT

It would appear that the AND function does not crash like the OR function but the end result is incorrect. The OR function simply won't graph due to a divide by 0 error. The NOT function appears to still be learnable. 

If desired, I can post my python script with how I generated these graphs. I am using a sigmoid activate function.
If you don't have a bias, your perceptron learns the function

$$y = w_1 \cdot x$$

This means the decision boundary is a line through (0, 0).

AND

$y, x, w_1 \in \mathbb{R}^2$

Function table:

x_1 x_2 y
---------
0   0   0
0   1   0
1   0   0
1   1   1


This is not learnable.


  I am using a sigmoid activate function.


Without a hidden layer, a Perceptron is still a linear classifier.