Finding variable values to make equations behave like boolean gate I came across following problem:

Consider the following simple perceptron, for which the inputs are the always $1$ feature and two binary features $x_1 ∈ \{0, 1\}$ and $x_2 ∈ \{0, 1\}$. The output $y ∈ \{0, 1\}$.
$$y=\begin{cases}
1,  &amp; \text{if $w_0+w_1.x_1+w_2.x_2&gt;0$ } \\
0, &amp; \text{otherwise}
\end{cases}$$
Choose all the correct option below that the weights $w_0,w_1,w_2$ should satisfy such that $y=(x_1 \text{ AND } x_2 )$.
(A) $w_0&gt;0$      (b) $w_0≤0$   (c) $w_1&gt;0$        (d) $w_2&gt;0$

I tried to solve the problem and came up with following:


(Eqn# stands for Equation# in above table)
Eqn#4 $= w_0≤0$  (Option B)
Eqn#2 $→w_0≤-w_1$    (Eqn#5)
Eqn#3 $→w_0≤-w_2$    (Eqn#6)
Eqn#1 $→ 0&gt;-w_1-w_2-w_0$
$→ 0&gt;-w_1+(non-ve)$    … from Eqn#3
$→ -(non-ve)&gt;-w_1$
$→ -ve&gt;-w_1$
$→ w_1&gt;0$ (Option C)
Similarly from Equation 4 and 2, it can be proven that $w_2&gt;0$ (Option D)

But I feel something is wrong with above. I feel option B contradicts with both option C and option D. For example, option C and Eqn#5 makes $w_0=0$ impossible which is allowed by option B. Where did I made mistake? I feel its incorrect to directly conclude $w_0\leq 0$ from Eqn#4 and &quot;stop&quot;. What I mean is that I have to continuously keep checking if all facts that I conclude lead to desired effect &quot;together&quot;. For example, once I got that option (c) is correct, I have to combine Option C fact with Eqn#4 fact to conclude that $w_0&lt;0$. So finding solution to such problems is kind of continuous process till we get all equations which satisfy each other and target goal (AND gate here). Q1. Am I correct with this? Or I am unnecessarily making things difficult and there is simpler and more importantly &quot;straight forward&quot; way (not involving continuously checking if all equations satisfy each other) to find such solutions?
Q2. This does also mean that Option B should have $&lt;$ instead of $\leq$. Am I correct with this too?
PS:
After getting confused like me, some of my friends have started saying it should be $\geq$ instead of $\gt$:
$$y=\begin{cases}
1,  &amp; \text{if $w_0+w_1.x_1+w_2.x_2\color{red}{\geq}0$ } \\
0, &amp; \text{otherwise}
\end{cases}$$
But I dont feel so as I can still find correct answer with given problem with conclusion that option B should have $&lt;$ instead of $\leq$. Also I dont feel its correct for a function to interpret $0$ as $True$. Q3. Am I correct with this too? (problem doesnt need correction, but option B)
You can only have $w_0 &lt; 0$. If $w_0 = 0$, then eq 1,2,3 imply that
$$ 0 &lt; w_{1,2} \leq 0 $$
which is a contradiction. Hence option b) is incorrect.