Let's break down the solution step by step. ### Step 1: **Identify the Equations** The image does not provide the explicit equations for outputs \(X\), \(Y\), and \(Z\). To solve this, **please provide the equations** for outputs \(X\), \(Y\), and \(Z\) in terms of inputs \(A\), \(B\), \(C\), and \(D\). --- #### **Assuming Hypothetical Equations** Let’s assume the following equations for the sake of demonstration (please correct them if you have the actual ones): - \( X = A \overline{B}C + B\overline{C}D \) - \( Y = AB + CD \) - \( Z = \overline{A}B + C\overline{D} \) --- ### Step 2: **Draw the PLA Connections** **A PLA (Programmable Logic Array)** consists of: - **Input lines** (here: A, B, C, D) and their complements (\(\overline{A}\), \(\overline{B}\), etc.) - **AND array**: To generate product (minterm) terms. - **OR array**: To sum those product terms for each output. #### **AND Array Connections** For each product term in the equations, **connect the required input or its complement**. **For \(X\):** - \(A\overline{B}C\): Connect \(A\), \(\overline{B}\), and \(C\). - \(B\overline{C}D\): Connect \(B\), \(\overline{C}\), and \(D\). **For \(Y\):** - \(AB\): Connect \(A\) and \(B\). - \(CD\): Connect \(C\) and \(D\). **For \(Z\):** - \(\overline{A}B\): Connect \(\overline{A}\) and \(B\). - \(C\overline{D}\): Connect \(C\) and \(\overline{D}\). **Mark the intersections on the AND plane for these connections.** #### **OR Array Connections** Each output line (X, Y, Z) will be connected to the required AND gates (product terms). --- ### Step 3: **Truth Table for ROM Implementation** A ROM implementation needs a **truth table** mapping every possible input combination to the outputs (\(X, Y, Z\)). #### **Inputs:** \(A, B, C, D\) (4 inputs \(\rightarrow\) 16 combinations) #### **Outputs:** \(X, Y, Z\) | A | B | C | D | X | Y | Z | |---|---|---|---|---|---|---| | 0 | 0 | 0 | 0 | ? | ? | ? | | 0 | 0 | 0 | 1 | ? | ? | ? | | 0 | 0 | 1 | 0 | ? | ? | ? | | 0 | 0 | 1 | 1 | ? | ? | ? | | 0 | 1 | 0 | 0 | ? | ? | ? | | 0 | 1 | 0 | 1 | ? | ? | ? | | 0 | 1 | 1 | 0 | ? | ? | ? | | 0 | 1 | 1 | 1 | ? | ? | ? | | 1 | 0 | 0 | 0 | ? | ? | ? | | 1 | 0 | 0 | 1 | ? | ? | ? | | 1 | 0 | 1 | 0 | ? | ? | ? | | 1 | 0 | 1 | 1 | ? | ? | ? | | 1 | 1 | 0 | 0 | ? | ? | ? | | 1 | 1 | 0 | 1 | ? | ? | ? | | 1 | 1 | 1 | 0 | ? | ? | ? | | 1 | 1 | 1 | 1 | ? | ? | ? | Fill in each row by substituting the values into the equations for X, Y, and Z. --- ### **Final Answer (General Steps):** #### **(a) PLA Connections:** 1. **For each product term in output equations, connect the corresponding input lines (or their complements) to the AND gates.** 2. **Connect the outputs of AND gates (product terms) to the OR gates for each output as required by the equations.** #### **(b) ROM Truth Table:** 1. **List all possible input combinations (here, 16 for 4 inputs).** 2. **Compute the outputs for each combination using the given equations and fill the table.** --- **If you provide the actual equations**, I can fill in the exact connections and complete the truth table for you!

Let's proceed with a detailed step-by-step solution assuming the problem involves designing a Programmable Logic Array (PLA) to implement given Boolean functions. Since the image is not explicitly providing the equations, I will assume typical example functions to illustrate the process. Please replace these with the actual functions you have for precise results. --- ### **Step 1: Understand the problem** - Inputs: \(A, B, C, D\) - Outputs: \(X, Y, Z\) - Goal: Use a PLA to implement the functions for \(X, Y, Z\) --- ### **Step 2: Determine the Boolean functions** Suppose the functions are: \[ \begin{cases} X = A \overline{B} C + B \overline{C} D \\ Y = AB + CD \\ Z = \overline{A} B + C \overline{D} \end{cases} \] **Note:** Replace these with your actual functions. --- ### **Step 3: List all minterms (truth table)** Create a truth table with all 16 input combinations (\(A, B, C, D\)), compute outputs for each function. | A | B | C | D | \(\overline{A}\) | \(\overline{B}\) | \(\overline{C}\) | \(\overline{D}\) | \(X\) | \(Y\) | \(Z\) | |---|---|---|---|-----------------|-----------------|-----------------|-----------------|-------|-------|-------| | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | ? | ? | ? | | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | ? | ? | ? | | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | ? | ? | ? | | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | ? | ? | ? | | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 1 | ? | ? | ? | | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | ? | ? | ? | | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 1 | ? | ? | ? | | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | ? | ? | ? | | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | ? | ? | ? | | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | ? | ? | ? | | 1 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | ? | ? | ? | | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | ? | ? | ? | | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | ? | ? | ? | | 1 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | ? | ? | ? | | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | ? | ? | ? | | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | ? | ? | ? | --- ### **Step 4: Compute outputs for each row** Let's analyze row by row. **Row 1:** \(A=0, B=0, C=0, D=0\) \[ \begin{cases} X = A \overline{B} C + B \overline{C} D = 0 \times 1 \times 0 + 0 \times 1 \times 0 = 0 + 0 = 0 \\ Y = AB + CD = 0 \times 0 + 0 \times 0 = 0 + 0 = 0 \\ Z = \overline{A} B + C \overline{D} = 1 \times 0 + 0 \times 1 = 0 + 0 = 0 \end{cases} \] Similarly, proceed for all rows. --- ### **Step 5: Summarize the output for all rows** | A | B | C | D | \(X\) | \(Y\) | \(Z\) | |---|---|---|---|--------|--------|--------| | 0 | 0 | 0 | 0 | 0 | 0 | 0 | | 0 | 0 | 0 | 1 | 0 | 0 | 0 | | 0 | 0 | 1 | 0 | 0 | 0 | 0 | | 0 | 0 | 1 | 1 | 0 | 0 | 0 | | 0 | 1 | 0 | 0 | 0 | 0 | 1 | | 0 | 1 | 0 | 1 | 0 | 0 | 1 | | 0 | 1 | 1 | 0 | 0 | 1 | 1 | | 0 | 1 | 1 | 1 | 0 | 1 | 1 | | 1 | 0 | 0 | 0 | 0 | 0 | 0 | | 1 | 0 | 0 | 1 | 0 | 0 | 0 | | 1 | 0 | 1 | 0 | 1 | 0 | 0 | | 1 | 0 | 1 | 1 | 1 | 0 | 0 | | 1 | 1 | 0 | 0 | 0 | 1 | 1 | | 1 | 1 | 0 | 1 | 0 | 1 | 1 | | 1 | 1 | 1 | 0 | 0 | 1 | 1 | | 1 | 1 | 1 | 1 | 0 | 1 | 1 | --- ### **Step 6: Identify the product terms (minterms)** From the functions, extract the minterms: - \(X = A \overline{B} C + B \overline{C} D\) - \(A \overline{B} C\): minterms where \(A=1, B=0, C=1\) - Rows: 11 (A=1, B=0, C=1, D=1) — output \(X=1\) - \(B \overline{C} D\): \(B=1, C=0, D=1\) - Row: 13 (A=1, B=1, C=0, D=1) - \(Y = AB + CD\) - \(AB\): \(A=1, B=1\) - Rows: 12, 13, 14, 15 - \(CD\): \(C=1, D=1\) - Rows: 3, 4, 7, 8 - \(Z = \overline{A} B + C \overline{D}\) - \(\overline{A} B\): \(A=0, B=1\) - Rows: 4, 6, 7, 8 - \(C \overline{D}\): \(C=1, D=0\) - Rows: 3, 11, 14, 15 --- ### **Step 7: Draw the PLA connections** For each minterm: - Connect the input lines (or their complements) to AND gates as needed: | Minterm | Inputs connected | Corresponding AND gate | |----------|---------------------|------------------------| | \(A \overline{B} C\) | \(A\), \(\overline{B}\), \(C\) | AND gate 1 | | \(B \overline{C} D\) | \(B\), \(\overline{C}\), \(D\) | AND gate 2 | | \(AB\) | \(A\), \(B\) | AND gate 3 | | \(CD\) | \(C\), \(D\) | AND gate 4 | | \(\overline{A} B\) | \(\overline{A}\), \(B\) | AND gate 5 | | \(C \overline{D}\) | \(C\), \(\overline{D}\) | AND gate 6 | - Connect each AND gate output to the OR gate for each function: | Output | OR gates (product terms) connected | |---------|-------------------------------------| | \(X\) | AND gate 1, AND gate 2 | | \(Y\) | AND gate 3, AND gate 4 | | \(Z\) | AND gate 5, AND gate 6 | --- ### **Step 8: Final implementation** - **Input lines:** \(A, B, C, D\), with their complements generated. - **AND array:** Generate product terms as per above. - **OR array:** Sum the product terms for each output. --- ### **Summary:** 1. **Identify the Boolean functions.** 2. **Express each function as a sum of minterms or product terms.** 3. **Map each minterm to AND gates with inputs/complements.** 4. **Connect AND gates to OR gates corresponding to each output.** 5. **Construct the truth table for the ROM if implementing via ROM.** --- ### **Final notes:** - To implement your specific functions, repeat this process using your exact equations. - For detailed wiring, draw the input lines, their complements, connect to AND gates, then connect AND gate outputs to OR gates for each output. --- **If you provide the exact equations, I can generate the precise connections and a complete truth table.**