---
title: "Minimal SOP Σm(3,4,6,8,10,15)+d, NOR-only"
subject: "Digital Logic"
chapter: 3
key: "semester_3/digital_logic:kmap-minimal-sop-nor"
source: https://www.bctengineer.com/q/semester_3/digital_logic/kmap-minimal-sop-nor
site: BCT Engineering
---

# Minimal SOP Σm(3,4,6,8,10,15)+d, NOR-only

Digital Logic · Chapter 3: Combinational Logic Circuits

- Asked in: 2078 Bhadra, 3+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-minimal-sop-nor
- Whole chapter: https://www.bctengineer.com/md/semester_3/digital_logic/chapter-3

## Question

Obtain the minimal SOP form of F(A,B,C,D) = Σm(3,4,6,8,10,15) + d(0,2,7,14) using K-map and implement the simplified result using NOR gate only.

## Short answer

The minimal SOP is F = A'C + A'D' + B'D' + BC. For NOR-only implementation, express each term as NOR of complemented literals (e.g., A'C = (A+C')'), combine with a NOR to get F', then invert with a NOR gate.

## Answer

Given $F(A,B,C,D) = \Sigma m(3,4,6,8,10,15) + d(0,2,7,14)$.

#### K-map and minimal SOP

```text
        CD
AB      00  01  11  10
      +---+---+---+---+
 00   | X | 0 | 1 | X |
      +---+---+---+---+
 01   | 1 | 0 | X | 1 |
      +---+---+---+---+
 11   | 0 | 0 | 1 | X |
      +---+---+---+---+
 10   | 1 | 0 | 0 | 1 |
      +---+---+---+---+
```

Every group below is a quad, made possible by the don't-cares 0, 2, 7, 14:

- Quad m(2,3,6,7) → $\overline{A}C$
- Quad m(0,2,4,6) → $\overline{A}\,\overline{D}$
- Quad m(0,2,8,10) → $\overline{B}\,\overline{D}$
- Quad m(6,7,14,15) → $BC$

$$
F = \overline{A}C + \overline{A}\,\overline{D} + \overline{B}\,\overline{D} + BC
$$

Check: 3 (and 2, 7) by $\overline{A}C$; 4, 6 by $\overline{A}\,\overline{D}$; 8, 10 by $\overline{B}\,\overline{D}$; 15 (with 7, 14) by $BC$. This is the only minimal SOP (4 terms, 8 literals).

#### Implementation using NOR gates only

For NOR-only realization of an SOP expression, write each product term as a NOR of complemented literals (De Morgan: $\overline{X}\,\overline{Y} = \overline{X+Y}$), feed these to a NOR gate (which gives $\overline{F}$), and invert with one more NOR whose inputs are tied.

$$
\begin{aligned} \overline{A}C &= \overline{A+\overline{C}} & \overline{A}\,\overline{D} &= \overline{A+D} \\ \overline{B}\,\overline{D} &= \overline{B+D} & BC &= \overline{\overline{B}+\overline{C}} \end{aligned}
$$
$$
F = \overline{\overline{P_1+P_2+P_3+P_4}}
$$

```text
B ──[NOR]── B'   (inputs tied)
C ──[NOR]── C'   (inputs tied)

A, C'  ──[NOR]── P1  ─┐
A, D   ──[NOR]── P2  ─┤
                      ├──[NOR]── F'
B, D   ──[NOR]── P3  ─┤
B', C' ──[NOR]── P4  ─┘

F' ──[NOR]── F   (inputs tied)
```

Gates: 2 NOR inverters ($\overline{B}, \overline{C}$), 4 two-input NORs, one 4-input NOR and one NOR inverter: 8 NOR gates.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Minimal SOP:
   A. A'C + A'D' + B'D' + BC
   B. AC + BD
   C. A'+B'
   D. CD+AB
   Answer: A. Four quads. (Key point: Result)

2. Role of don't cares:
   A. Count as 0
   B. Enlarge groups to quads
   C. Ignored
   D. Add terms
   Answer: B. 0,2,7,14 used. (Key point: Don't cares)

3. A'C as NOR:
   A. (A+C)'
   B. (A'+C)'
   C. (A + C')'
   D. (AC)'
   Answer: C. A'C=(A+C')'. (Key point: NOR product)

4. Total NOR gates:
   A. 6
   B. 7
   C. 10
   D. 8
   Answer: D. 2 inv + 4 + 1 + 1. (Key point: Gates)

## Related in Digital Logic

- [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 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
- [K-map Σm(0,1,2,8,10,14,15)+d, POS with NOR](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Akmap-pos-nor-0128) — Digital Logic, chapter 3
- [K-map Σm(0,2,4,6,8,10,15)+d, NOR](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Akmap-nor-reduced-0246) — Digital Logic, chapter 3
- [Minterms/maxterms; K-map Σm(0,1,2,5,8,14)+d, NOR](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Aminterm-maxterm-kmap-nor-0125) — Digital Logic, chapter 3

## Related in other subjects

- [K-map with don't cares, NAND only](https://www.bctengineer.com/question.md?key=semester_2%2Fbasic_electronics_engineering%3Akmap-dont-care-nand) — Basic Electronics Engineering, chapter 6