--- title: "K-map Σm(0,1,2,8,10,14,15)+d, POS with NOR" subject: "Digital Logic" chapter: 3 key: "semester_3/digital_logic:kmap-pos-nor-0128" source: https://www.bctengineer.com/q/semester_3/digital_logic/kmap-pos-nor-0128 site: BCT Engineering --- # K-map Σm(0,1,2,8,10,14,15)+d, POS with NOR Digital Logic · Chapter 3: Combinational Logic Circuits - Asked in: 2078 Bhadra, 4+3 marks - Video: [Minimize the following expression using K map and realize using NOR gate](https://www.youtube.com/watch?v=2HpvaRRDPK8) — Maharshi Sanand Yadav T - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/digital_logic/kmap-pos-nor-0128 - Whole chapter: https://www.bctengineer.com/md/semester_3/digital_logic/chapter-3 ## Question Simplify Σm(0,1,2,8,10,14,15) d = (3,7,11,13) using k-map, write its standard product of sum expression and realize it using NOR gates only. ## Short answer Simplified POS: F = (A+B')(B'+C)(A'+B+D'). Standard POS: ΠM(4,5,6,7,9,11,12,13). Realized with NOR gates using NOR-NOR form and inverters. ## Answer Given $F(A,B,C,D) = \Sigma m(0,1,2,8,10,14,15) + d(3,7,11,13)$. The 0-cells are 4, 5, 6, 9, 12. #### K-map (grouping the 0s for POS) ```text CD AB 00 01 11 10 +---+---+---+---+ 00 | 1 | 1 | X | 1 | +---+---+---+---+ 01 | 0 | 0 | X | 0 | +---+---+---+---+ 11 | 0 | X | 1 | 1 | +---+---+---+---+ 10 | 1 | 0 | X | 1 | +---+---+---+---+ ``` - Quad of 0s M(4,5,6,7) → $(A+\overline{B})$ - Quad of 0s M(4,5,12,13) → $(\overline{B}+C)$ - Pair of 0s M(9,11) → $(\overline{A}+B+\overline{D})$ $$ F = (A+\overline{B})(\overline{B}+C)(\overline{A}+B+\overline{D}) $$ Check: 4, 5, 6 by $(A+\overline{B})$; 12 by $(\overline{B}+C)$; 9 by $(\overline{A}+B+\overline{D})$. ($(\overline{A}+C+\overline{D})$, cells 9 and 13, is an equally good third term.) #### Standard product of sums The given 0s are $\Pi M(4,5,6,9,12)$ with don't-cares $d(3,7,11,13)$. The simplified POS also makes don't-cares 7, 11, 13 equal to 0, so its standard (canonical) POS is: $$ F = \Pi M(4,5,6,7,9,11,12,13) $$ $$ \begin{aligned} F = {} & (A+\overline{B}+C+D)(A+\overline{B}+C+\overline{D})(A+\overline{B}+\overline{C}+D)(A+\overline{B}+\overline{C}+\overline{D}) \\ & (\overline{A}+B+C+\overline{D})(\overline{A}+B+\overline{C}+\overline{D})(\overline{A}+\overline{B}+C+D)(\overline{A}+\overline{B}+C+\overline{D}) \end{aligned} $$ #### Realization using NOR gates only A POS expression maps directly to a two-level NOR-NOR circuit: $$ F = \overline{\overline{(A+\overline{B})} + \overline{(\overline{B}+C)} + \overline{(\overline{A}+B+\overline{D})}} $$ ```text A ──[NOR]── A' (inputs tied) B ──[NOR]── B' (inputs tied) D ──[NOR]── D' (inputs tied) A, B' ──[NOR]── P1 ─┐ B', C ──[NOR]── P2 ─┤──[NOR]── F A', B, D' ──[NOR]── P3 ─┘ ``` Gates: 3 NOR inverters ($\overline{A}, \overline{B}, \overline{D}$), two 2-input NORs, one 3-input NOR and one 3-input output NOR: 7 NOR gates. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Which quad of 0s gives (A + B')? A. M(4,5,6,7) B. M(4,5,12,13) C. M(0,1,2,3) D. M(8,9,10,11) Answer: A. Cells 4-7 have A=0 and B=1; a group of 0s gives the sum A + B'. (Key point: Groups of 0s) 2. What is the simplified POS? A. (A+B')(B'+C)(A'+B+D') B. (A+B)(B+C')(A'+B'+D) C. (A'+B)(C'+D) D. (A+B')(C+D) Answer: A. Quads M(4-7) = A+B', M(4,5,12,13) = B'+C and pair M(9,11) = A'+B+D'. (Key point: Simplified POS) 3. Which is the standard POS list? A. ΠM(4,5,6,9,12) B. ΠM(4,5,6,7,9,11,12,13) C. ΠM(3,7,11,13) D. ΠM(0,1,2,8,10,14,15) Answer: B. Don't-cares 7, 11 and 13 are taken as 0 by the groups, adding them to the original 0s. (Key point: Standard POS) 4. How many NOR gates are used? A. 4 B. 5 C. 7 D. 9 Answer: C. 3 inverters + two 2-input NORs + one 3-input NOR + one 3-input output NOR = 7. (Key point: NOR-only gates) ## Related in Digital Logic - [K-map Σ(1,2,3,8,9,10,11,13,14)+d, POS](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Akmap-pos-form-12389) — Digital Logic, chapter 3 - [K-map Σm(7,9,12,13,14,15)+d, NOR-only](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Akmap-nor-sum-7-9-12) — Digital Logic, chapter 3 - [K-map Σm(0,1,2,4,7,8,9,10,12,15)+d, NOR-only](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Akmap-nor-only-01247) — Digital Logic, chapter 3 - [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 Σ(0,1,4,8,10,11,12)+d, NOR](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Akmap-nor-gates-01248) — Digital Logic, chapter 3, asked 5× - [Minterms/maxterms; K-map Σm(2,4,5,13,14)+d, universal gates](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Aminterm-maxterm-kmap-universal) — Digital Logic, chapter 3