Deep Learning · TEE · section

MT1B Q2

Current MT1 and the two teacher backprop examples

Question

Question (3 marks). Show how a perceptron with w₁ = w₂ = 0.6 and bias −0.8 implements AND.

Answer — step by step

Given

Binary inputs 00,01,10,11.

w₁ = w₂ = 0.6; b = −0.8; output 1 when net ≥ 0.

To find

Calculate all four net values and outputs to show AND.

Formula — which one and why

Threshold logic. Compute the weighted sum for every binary input combination and apply the threshold rule exactly. Hidden neurons can provide separate threshold decisions before the output combines them.

step(z) returns 1 at or above zero unless the question specifies a strict inequality; otherwise it returns 0.

$z=\sum_iw_ix_i+b$ LaTeX: z=\sum_iw_ix_i+b

$\operatorname{step}(z)=\begin{cases}1&z\ge0\\0&z<0\end{cases}$ LaTeX: \operatorname{step}(z)=\begin{cases}1&z\ge0\\0&z<0\end{cases}

Yaad rakhne ka cue: Har input row ka sum nikalo, phir threshold lagao.

Solution — step by step

Input (0, 0)

Step 1: multiply input 1 by its weight.

$0.6(0)=0$ LaTeX: 0.6(0)=0

Step 2: multiply input 2 by its weight.

$0.6(0)=0$ LaTeX: 0.6(0)=0

Step 3: add the two products.

$s=0+0=0$ LaTeX: s=0+0=0

Step 4: add the bias once.

$z=0+(-0.8)=-0.8$ LaTeX: z=0+(-0.8)=-0.8

Step 5: apply the stated threshold.

z = -0.8; output = 0 because output is 1 exactly when z ≥ 0.

Input (0, 1)

Step 1: multiply input 1 by its weight.

$0.6(0)=0$ LaTeX: 0.6(0)=0

Step 2: multiply input 2 by its weight.

$0.6(1)=0.6$ LaTeX: 0.6(1)=0.6

Step 3: add the two products.

$s=0+0.6=0.6$ LaTeX: s=0+0.6=0.6

Step 4: add the bias once.

$z=0.6+(-0.8)=-0.2$ LaTeX: z=0.6+(-0.8)=-0.2

Step 5: apply the stated threshold.

z = -0.2; output = 0 because output is 1 exactly when z ≥ 0.

Input (1, 0)

Step 1: multiply input 1 by its weight.

$0.6(1)=0.6$ LaTeX: 0.6(1)=0.6

Step 2: multiply input 2 by its weight.

$0.6(0)=0$ LaTeX: 0.6(0)=0

Step 3: add the two products.

$s=0.6+0=0.6$ LaTeX: s=0.6+0=0.6

Step 4: add the bias once.

$z=0.6+(-0.8)=-0.2$ LaTeX: z=0.6+(-0.8)=-0.2

Step 5: apply the stated threshold.

z = -0.2; output = 0 because output is 1 exactly when z ≥ 0.

Input (1, 1)

Step 1: multiply input 1 by its weight.

$0.6(1)=0.6$ LaTeX: 0.6(1)=0.6

Step 2: multiply input 2 by its weight.

$0.6(1)=0.6$ LaTeX: 0.6(1)=0.6

Step 3: add the two products.

$s=0.6+0.6=1.2$ LaTeX: s=0.6+0.6=1.2

Step 4: add the bias once.

$z=1.2+(-0.8)=0.4$ LaTeX: z=1.2+(-0.8)=0.4

Step 5: apply the stated threshold.

z = 0.4; output = 1 because output is 1 exactly when z ≥ 0.

Final answer: outputs are [0, 0, 0, 1] for inputs 00, 01, 10, 11: exactly AND.