# Solution to Question 1 Given function in canonical form: \[ F = \sum m(,2,6,7,8,9,11,13,18,22,25,28,30) + \sum d(1,4,5,17,19,31) \] where \( \sum m \) are minterms and \( \sum d \) are don't cares. Let's proceed step-by-step: --- ## **Step 1: Identify Number of Variables** The highest index in the minterms/don't cares is 31, which means **5 variables** (\(2^5 = 32\)): Let variables be: \( A, B, C, D, E \) ( = MSB, E = LSB). --- ## 2: List Out Minterms and Don't Cares** - **Minterms:** , 2, 6, 7, 8, 9, 11, 13, 18, 22, 25, 28, 30 - **Don't Cares:** 1, 4, 5, 17, 19, 31 --- ## **Step 3: Draw the First Order Variable Entered Map (VEM)** Let's use **A** as the first-order entered variable. We'll create a 4-variable K-map for \(B, C, D, E\), with two maps: one for \(A=\) and one for \(A=1\). ### **Step 3a: Fill the K-Map** #### For \(A=\) (minterms –15): - Minterms: , 2, 6, 7, 8, 9, 11, 13 - Don't Cares: 1, 4, 5 #### For \(A=1\) (minterms 16–31): - Minterms: 18, 22, 25, 28, 30 - Don't Cares: 17, 19, 31 --- ### **Step 3b: Draw K-Maps** #### Map for \(A=\) (\(m(\) to \(m(15)\)): | | 00 | 01 | 11 | 10 | |------|----|----|----|----| | **00** | 1 | DC | | 1 | | **01** | DC | | 1 | DC | | **11** | | 1 | 1 | | | **10** | 1 | 1 | | | #### Map for \(A=1\) (\(m(16)\) to \(m(31)\)): | | 00 | 01 | 11 | 10 | |------|----|----|----|----| | **00** | | DC | | | | **01** | DC | 1 | | DC | | **11** | | 1 | DC | | | **10** | 1 | | 1 | | (**Fill in each cell using binary representation; let me know if you need full K-map filling details.**) --- ## **Step 4: Simplify Using Groups** ### **For \(A=\):** Find largest groups (use don't cares where possible): - **Group 1:** (m6, m7, m14, m15): \(B=1, D=1\) for \(A=\) - **Group 2:** (m2, m3, m6, m7): \(C=1, D=1\) for \(A=\) - **Group 3:** (m8, m9, m10, m11): \(B=, C=\) for \(A=\) ### **For \(A=1\):** - **Group 1:** (m22, m23, m30, m31): \(C=1, E=\) for \(A=1\) - **Group 2:** (m25, m24, m28, m29): \(B=, D=\) for \(A=1\) --- ## **Step 5: Write Simplified Expression** Combine the groups: \[ F = A' [B D + C D + B' C'] + A [C E' + B' D'] \] --- ## **Step 6: Implement Using NAND Gates Only** Convert each AND, OR, and NOT into NAND equivalents. ### **Step 6a: Express in terms of AND, OR, NOT** \[ F = A' [(B D) + (C D) + (B' C')] + A [(C E') + (B' D')] \] ### **Step 6b: Implement NAND equivalents** - \(X \cdot Y = (X \text{ NAND } Y)'\) (use double NAND for AND) - \(X + Y = (X' \cdot Y')' = (X \text{ NAND } X) \text{ NAND } (Y \text{ NAND } Y)\) - \(X' = X \text{ NAND } X\) #### **Rewrite** Let’s convert each term into NAND logic. --- ## **Final Simplified Expression** \[ \boxed{F = A' [B D + C D + B' C'] + A [C E' + B' D']} \] ### **NAND Implementation (Block Diagram Suggestion)** 1. **Inverters:** (using NAND gates) - \(A' = A \text{ NAND } A\) - \(B' = B \text{ NAND } B\) - \(C' = C \text{ NAND } C\) - \(E' = E \text{ NAND } E\) - \(D' = D \text{ NAND } D\) 2. **ANDs:** (using NAND gates) - \(B D = (B \text{ NAND } D)'\) - \(C D = (C \text{ NAND } D)'\) - \(B' C' = (B' \text{ NAND } C')'\) - \(C E' = (C \text{ NAND } E')'\) - \(B' D' = (B' \text{ NAND } D')'\) 3. **ORs:** (using NAND gates) - \([B D + C D + B' C'] = ((B D)' \text{ NAND } (C D)') \text{ NAND } ((B' C')')\) - \([C E' + B' D'] = ((C E')' \text{ NAND } (B' D')')\) 4. **Combine with A and A':** - \(A' \text{ AND } [B D + C D + B' C']\) - \(A \text{ AND } [C E' + B' D']\) 5. **Final OR:** - Combine outputs from steps above using NAND-based OR. --- ### **Summary Table** | Step | Logic Expression | NAND Equivalent | |------|------------------|----------------| | NOT | \(X'\) | \(X \text{ NAND } X\) | | AND | \(X \cdot Y\) | \((X \text{ NAND } Y) \text{ NAND } (X \text{ NAND } Y)\) | | OR | \(X + Y\) | \((X' \text{ NAND } Y')\) | --- ## **Final Answers** ### **a) Simplified Function** \[ \boxed{F = A' [B D + C D + B' C'] + A [C E' + B' D']} \] ### **b) Implement using NAND gates only** - Use the above logic, replacing every NOT, AND, OR with NAND equivalents as shown. - Draw the circuit as a block diagram with each step in NAND logic (details as above). --- **If you’d like a full step-by-step K-map filling and exact NAND gate diagram, let me know!**