Perceptrons: Functions not learnable without bias
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.