--- title: "K-map Σ(1,2,3,8,10,13)+d, standard SOP" subject: "Digital Logic" chapter: 3 key: "semester_3/digital_logic:kmap-standard-sop-123810" source: https://www.bctengineer.com/q/semester_3/digital_logic/kmap-standard-sop-123810 site: BCT Engineering --- # K-map Σ(1,2,3,8,10,13)+d, standard SOP Digital Logic · Chapter 3: Combinational Logic Circuits - Asked in: 2073 Shrawan, 6 marks - Video: [Don't care conditions in Digital Electronics | SOP and POS Kmap with don't care conditions](https://www.youtube.com/watch?v=wduY7oZnF-g) — Electronics Subjectified - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/digital_logic/kmap-standard-sop-123810 - Whole chapter: https://www.bctengineer.com/md/semester_3/digital_logic/chapter-3 ## Question Simplify Σ(1,2,3,8,10,13) + d(0,4,5,6,7,9,12) by using K-Map and write its standard SOP expression. ## Short answer Using K-map with don’t-cares taken as 1, the simplified expression is F = A' + C' + B' D'. The standard SOP (canonical) expands this to include all minterms where F=1, giving Σm(0,1,2,3,4,5,6,7,8,9,10,12,13). ## Answer Given $F(A,B,C,D) = \Sigma m(1,2,3,8,10,13) + d(0,4,5,6,7,9,12)$. #### K-map simplification ```text CD AB 00 01 11 10 +---+---+---+---+ 00 | X | 1 | 1 | 1 | +---+---+---+---+ 01 | X | X | X | X | +---+---+---+---+ 11 | X | 1 | 0 | 0 | +---+---+---+---+ 10 | 1 | X | 0 | 1 | +---+---+---+---+ ``` The many don't-cares allow three large groups (two octets and one quad): - Octet m(0,1,2,3,4,5,6,7) → $\overline{A}$ - Octet m(0,1,4,5,8,9,12,13) → $\overline{C}$ - Quad m(0,2,8,10) → $\overline{B}\,\overline{D}$ $$ F = \overline{A} + \overline{C} + \overline{B}\,\overline{D} $$ Check: 1, 2, 3 by $\overline{A}$; 8, 13 (and 1) by $\overline{C}$; 10 (and 2, 8) by $\overline{B}\,\overline{D}$. All six minterms are covered; don't-cares 0, 4, 5, 6, 7, 9, 12 are taken as 1, so the 0s are only 11, 14, 15. #### Standard SOP expression The standard (canonical) SOP lists every minterm where the simplified F is 1. Expanding: - $\overline{A}$ → $m_0$ to $m_7$ - $\overline{C}$ → $m_0, m_1, m_4, m_5, m_8, m_9, m_{12}, m_{13}$ - $\overline{B}\,\overline{D}$ → $m_0, m_2, m_8, m_{10}$ $$ F = \Sigma m(0,1,2,3,4,5,6,7,8,9,10,12,13) $$ $$ \begin{aligned} F = {} & \overline{A}\,\overline{B}\,\overline{C}\,\overline{D} + \overline{A}\,\overline{B}\,\overline{C}D + \overline{A}\,\overline{B}C\overline{D} + \overline{A}\,\overline{B}CD + \overline{A}B\overline{C}\,\overline{D} + \overline{A}B\overline{C}D + \overline{A}BC\overline{D} \\ & + \overline{A}BCD + A\overline{B}\,\overline{C}\,\overline{D} + A\overline{B}\,\overline{C}D + A\overline{B}C\overline{D} + AB\overline{C}\,\overline{D} + AB\overline{C}D \end{aligned} $$ Without the don't-cares, the required minterms alone are $\Sigma m(1,2,3,8,10,13)$. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Which groups form the minimal cover of Σ(1,2,3,8,10,13) + d(0,4,5,6,7,9,12)? A. A', C' and B'D' B. A', B' and C'D' C. A'C and AC' D. A', C and B'D' Answer: A. Octets m(0-7) = A' and m(0,1,4,5,8,9,12,13) = C', plus quad m(0,2,8,10) = B'D'. (Key point: Large groups with don't-cares) 2. What is the simplified F? A. A' + C' + B'D' B. A' + C + BD C. A'C' + B'D' D. A' + C' Answer: A. F = A' + C' + B'D'; the quad B'D' is required to cover minterm 10. (Key point: Simplified result) 3. What does the 'standard SOP' list? A. Only the original minterms 1,2,3,8,10,13 B. Every minterm where the simplified F is 1, including don't-cares set to 1 C. Only the don't-care cells D. The simplified product terms only Answer: B. The standard SOP expands the simplified F into all 4-variable minterms where it equals 1, so don't-cares used as 1 appear. (Key point: Standard SOP) 4. Which is the standard SOP minterm list of the simplified F? A. Σm(1,2,3,8,10,13) B. Σm(0,1,2,3,4,5,6,7,8,9,10,12,13) C. Σm(0,1,2,3,4,5,6,7) D. Σm(11,14,15) Answer: B. Expanding A', C' and B'D' gives 13 minterms; the 0s are only 11, 14 and 15. 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