Let's break down the problem step by step: ### **Given:** Function \( f(w, x, y, z) = \sum m(2, 5, 6, 7, 10, 15) \) You are to: - Determine the **minimum POS** (Product of Sums) using K-Map. - Draw a **minimum SOP** (Sum of Products) logic circuit. --- ## **Step 1: Write the Minterms in Binary** List out each minterm for 4 variables (w, x, y, z): | Decimal | Binary | w | x | y | z | |---------|--------|---|---|---|---| | 2 | 0010 | 0 | 0 | 1 | 0 | | 5 | 0101 | 0 | 1 | 0 | 1 | | 6 | 0110 | 0 | 1 | 1 | 0 | | 7 | 0111 | 0 | 1 | 1 | 1 | | 10 | 1010 | 1 | 0 | 1 | 0 | | 15 | 1111 | 1 | 1 | 1 | 1 | --- ## **Step 2: Draw the 4-variable K-Map** Let's draw a K-Map and mark **1s** in the minterm positions. Let’s use (w, x) for rows and (y, z) for columns: | | yz=00 | yz=01 | yz=11 | yz=10 | |------|-------|-------|-------|-------| | wx=00| 0 | 0 | 0 | 1 | (2) | wx=01| 0 | 1 | 1 | 1 | (5, 7, 6) | wx=11| 0 | 0 | 1 | 0 | (15) | wx=10| 0 | 0 | 0 | 1 | (10) --- ## **Step 3: Fill the K-Map** ``` yz 00 01 11 10 wx --------------------- 00 | 0 0 0 1 (2) 01 | 0 1 1 1 (5, 7, 6) 11 | 0 0 1 0 (15) 10 | 0 0 0 1 (10) ``` --- ## **Step 4: Find the Minimum POS Expression** To get **POS**, group the 0s (since POS = product of maxterms). ### **Maxterms are the places where the output is 0.** Write the maxterm indices (cells with 0): - 0, 1, 3, 4, 8, 9, 11, 12, 13, 14 Let's write the maxterms: - (0): w' x' y' z' - (1): w' x' y' z - (3): w' x' y z - (4): w' x y' z' - (8): w x' y' z' - (9): w x' y' z - (11): w x' y z - (12): w x y' z' - (13): w x y' z - (14): w x y z' The corresponding maxterms for each cell are: - (w + x + y + z) - (w + x + y + z') - (w + x + y' + z) - (w + x' + y + z) - (w' + x + y + z) - (w' + x + y + z') - (w' + x + y' + z) - (w' + x' + y + z) - (w' + x' + y + z') - (w' + x' + y' + z) Now, group the 0s in the K-Map for simplification. --- ### **Grouping 0's in the K-Map** Let's visually group the 0's for the largest possible groups (groups of 8, 4, or 2): - Top-left 2x2 block (cells 0, 1, 4, 5) (but 5 is 1, so only 0,1,4,8,9): try to group as much as possible. Finally, after grouping and simplifying, the minimum POS expression is: #### **Minimum POS:** \[ (w + x + z)(w + x' + z')(w' + x + y) \] --- ## **Step 5: Minimum SOP Expression and Logic Circuit** To minimize the SOP, group the 1s in the K-map: - Group 1: (6, 7) → wx'y - Group 2: (5, 7) → w'x y z - Group 3: (2, 10) → x'y z' - Group 4: (15) → wx y z #### **Minimum SOP expression:** \[ f = wx'y + w'x y z + x'y z' + wx y z \] **Draw the logic circuit** for this expression using AND, OR gates according to the minimized SOP. --- ## **Final Answers** ### **Minimum POS (Product of Sums):** \[ \boxed{(w + x + z)(w + x' + z')(w' + x + y)} \] ### **Minimum SOP (Sum of Products):** \[ \boxed{wx'y + w'x y z + x'y z' + wx y z} \] **Draw the SOP circuit** using AND gates for each term and OR gate to sum them up. --- If you want the K-map illustration or circuit diagram, let me know!