🔹 AND Gate using Perceptron

1. 🧠 Problem Definition

The AND gate outputs 1 only when both inputs are 1.

Truth Table:

x₁	x₂	Output (y)
0	0	0
0	1	0
1	0	0
1	1	1

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2. ⚙️ Perceptron Model

A perceptron computes:

$y = \text{step}(w_1 x_1 + w_2 x_2 + b)$

Where:

• $w_1, w_2$: weights
• $b$: bias
• step(): activation function (0 or 1)

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3. 🔢 Choosing Weights and Bias

To implement AND, we need:

• Output = 1 only when both inputs are 1

A valid choice:

• $w_1 = 1$
  
• $w_2 = 1$
  
• $b = -1.5$
  

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4. 🧮 Verification

Let’s test all inputs:

Case 1: (0, 0)

$$1(0) + 1(0) - 1.5 = -1.5 \Rightarrow 0$$

Case 2: (0, 1)

$$1(0) + 1(1) - 1.5 = -0.5 \Rightarrow 0$$

Case 3: (1, 0)

$$1(1) + 1(0) - 1.5 = -0.5 \Rightarrow 0$$

Case 4: (1, 1)

$$1(1) + 1(1) - 1.5 = 0.5 \Rightarrow 1$$

✅ Matches the AND truth table perfectly.

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5. 📐 Geometric Interpretation

• The perceptron creates a decision boundary (line):

$$x_1 + x_2 = 1.5$$

• Points:
  • (0,0), (0,1), (1,0) → below the line → class 0
  • (1,1) → above the line → class 1

➡️ This shows AND is linearly separable.

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6. 🎯 Key Insight

The perceptron works for the AND gate because the data can be separated by a straight line.

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🔹 Summary

A single-layer perceptron implements an AND gate by choosing weights and bias such that only the input (1,1) produces an output above the threshold.