--- title: "K-map Π(0,4,5,8,9,11,15), standard SOP" subject: "Digital Logic" chapter: 3 key: "semester_3/digital_logic:kmap-maxterm-sop-0458" source: https://www.bctengineer.com/q/semester_3/digital_logic/kmap-maxterm-sop-0458 site: BCT Engineering --- # K-map Π(0,4,5,8,9,11,15), standard SOP Digital Logic · Chapter 3: Combinational Logic Circuits - Asked in: 2068 Baisakh, 4+2 marks - Video: [SOP AND POS WITH K-MAP - Minimize SOP and POS with K-map solved examples - Hindi](https://www.youtube.com/watch?v=i5QjUVgY-XU) — 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-maxterm-sop-0458 - Whole chapter: https://www.bctengineer.com/md/semester_3/digital_logic/chapter-3 ## Question Simplify Π(0,4,5,8,9,11,15) using K-Map and write its standard SOP expression. ## Short answer K-map simplification groups 1s to obtain F = A' C + C D' + A B C' + A' B' D. The standard SOP is the sum of all 1-cells: Σm(1,2,3,6,7,10,12,13,14). ## Answer Given $F(A,B,C,D) = \Pi M(0,4,5,8,9,11,15)$. These cells are 0; the remaining cells are 1: $$ F = \Sigma m(1,2,3,6,7,10,12,13,14) $$ #### K-map ```text CD AB 00 01 11 10 +---+---+---+---+ 00 | 0 | 1 | 1 | 1 | +---+---+---+---+ 01 | 0 | 0 | 1 | 1 | +---+---+---+---+ 11 | 1 | 1 | 0 | 1 | +---+---+---+---+ 10 | 0 | 0 | 0 | 1 | +---+---+---+---+ ``` **Grouping the 1s:** - Quad m(2,3,6,7) → $\overline{A}C$ - Quad m(2,6,10,14) → $C\overline{D}$ - Pair m(12,13) → $AB\overline{C}$ - Pair m(1,3) → $\overline{A}\,\overline{B}D$ $$ F = \overline{A}C + C\overline{D} + AB\overline{C} + \overline{A}\,\overline{B}D $$ Check: 2, 3, 6, 7 by $\overline{A}C$; 10, 14 (with 2, 6) by $C\overline{D}$; 12, 13 by $AB\overline{C}$; 1 (with 3) by $\overline{A}\,\overline{B}D$. (Grouping the 0s gives the POS form $F = (A+\overline{B}+C)(A+C+D)(\overline{A}+B+C)(\overline{A}+\overline{C}+\overline{D})$, which has more literals, so the SOP form is the simplified answer.) #### Standard SOP expression The standard (canonical) SOP has one 4-literal minterm for each 1-cell: $$ \begin{aligned} F = {} & \overline{A}\,\overline{B}\,\overline{C}D + \overline{A}\,\overline{B}C\overline{D} + \overline{A}\,\overline{B}CD + \overline{A}BC\overline{D} + \overline{A}BCD \\ & + A\overline{B}C\overline{D} + AB\overline{C}\,\overline{D} + AB\overline{C}D + ABC\overline{D} \end{aligned} $$ $$ = \Sigma m(1,2,3,6,7,10,12,13,14) $$ ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. For ΠM(0,4,5,8,9,11,15), how are the 1-cells found? A. They are the listed numbers B. They are all other numbers 1,2,3,6,7,10,12,13,14 C. They are the don't-cares D. Only 0 and 15 Answer: B. Maxterms are the 0-cells; the remaining numbers are the 1s. (Key point: Converting maxterms to 1-cells) 2. Which quad is formed by minterms 2,3,6,7? A. A'C B. C D' C. A'B D. BC Answer: A. Cells 2,3,6,7 have A=0 and C=1, giving A'C. (Key point: Quad grouping) 3. What is the simplified SOP? A. A'C + CD' + ABC' + A'B'D B. A'C + CD' + AB + A'D C. A'C + BD + AC' D. C + A'D Answer: A. Quads A'C and CD' plus pairs ABC' (12,13) and A'B'D (1,3). (Key point: Simplified SOP) 4. How many terms does the standard SOP of this function have? A. 5 B. 9 C. 7 D. 16 Answer: B. There are 9 one-cells, each giving a four-literal minterm. (Key point: Standard SOP) ## Close variants set in other exams - [K-map Σ(1,2,3,8,10,13)+d, standard SOP](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Akmap-standard-sop-123810) — Digital Logic, chapter 3, asked 2× ## Related in Digital Logic - [K-map Σ(2,3,6,7,9,10,11)+d in SOP and POS](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Akmap-sop-pos-2367) — Digital Logic, chapter 3 - [K-map Σ(0,1,4,8,10,11,12)+d, standard minterm](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Akmap-standard-minterm-01248) — Digital Logic, chapter 3 - [K-map F(M,N,O,P) in SOP using NAND](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Akmap-sop-nand-mnop) — Digital Logic, chapter 3 - [K-map Σ(1,3,7,11,15)+d, NAND](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Akmap-nand-1371115) — Digital Logic, chapter 3, asked 5× - [K-map ΠM(0,2,3,8,10,11,12,15)+d, NOR](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Akmap-maxterm-nor-0238) — Digital Logic, chapter 3 - [K-map Π(0,2,5,8,10)+d, SOP and NOR](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Akmap-maxterm-sop-nor-025) — Digital Logic, chapter 3 ## Related in other subjects - [K-map: 3-variable five-term expression](https://www.bctengineer.com/question.md?key=semester_2%2Fbasic_electronics_engineering%3Akmap-xyz-five-term) — Basic Electronics Engineering, chapter 6 - [Population, parameter, statistic; samples of 3,7,11,15](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Apopulation-parameter-statistic-3-7-11) — Probability and Statistics, chapter 4 - [Loss and drop, 15 km 11 kV feeder](https://www.bctengineer.com/question.md?key=semester_8%2Ftransmission_and_distribution_system_design%3Audl-feeder-15km-11kv) — Transmission and Distribution System Design, chapter 7, asked 2× - [K-map: 4-variable minterm list](https://www.bctengineer.com/question.md?key=semester_2%2Fbasic_electronics_engineering%3Akmap-4var-sigma) — Basic Electronics Engineering, chapter 6