1. **Problem Statement:** We are asked to use the backpropagation algorithm to develop and train a neural network given input-output pairs. 2. **Understanding Backpropagation:** Backpropagation is a supervised learning algorithm used for training artificial neural networks. It involves a forward pass to compute outputs and a backward pass to update weights by minimizing the error between predicted and actual outputs. 3. **Key Formulas:** - Forward pass output: $$y = f(\sum w_i x_i + b)$$ where $f$ is the activation function. - Error: $$E = \frac{1}{2} (y_{target} - y)^2$$ - Weight update rule: $$w_i := w_i - \eta \frac{\partial E}{\partial w_i}$$ where $\eta$ is the learning rate. 4. **Step-by-step Training:** - Initialize weights and bias randomly. - For each input-output pair: - Compute the output using the current weights. - Calculate the error. - Compute gradients of error w.r.t weights. - Update weights using the gradient descent rule. - Repeat for multiple epochs until error converges. 5. **Example with one input neuron and one output neuron:** - Let input $x$, weight $w$, bias $b$, activation function $f$ (e.g., sigmoid). - Forward pass: $$y = f(wx + b)$$ - Error: $$E = \frac{1}{2}(y_{target} - y)^2$$ - Backpropagation updates: - $$\frac{\partial E}{\partial w} = -(y_{target} - y) f'(wx + b) x$$ - $$\frac{\partial E}{\partial b} = -(y_{target} - y) f'(wx + b)$$ - Update weights: - $$w := w - \eta \frac{\partial E}{\partial w}$$ - $$b := b - \eta \frac{\partial E}{\partial b}$$ 6. **Training on the given data:** - Use the 12 input-output pairs. - Choose learning rate $\eta$ (e.g., 0.1). - Iterate over data multiple times, updating weights and bias each time. 7. **Result:** After sufficient training, the neural network will approximate the mapping from input to output. This process requires iterative computation and cannot be fully shown here step-by-step for all data points due to length, but the above outlines the method to develop and train the network using backpropagation.