---
title: "K-map Σm(0,1,2,4,7,8,9,10,12,15)+d, NOR-only"
subject: "Digital Logic"
chapter: 3
key: "semester_3/digital_logic:kmap-nor-only-01247"
source: https://www.bctengineer.com/q/semester_3/digital_logic/kmap-nor-only-01247
site: BCT Engineering
---

# K-map Σm(0,1,2,4,7,8,9,10,12,15)+d, NOR-only

Digital Logic · Chapter 3: Combinational Logic Circuits

- Asked in: 2080 Bhadra, 4+2 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-nor-only-01247
- Whole chapter: https://www.bctengineer.com/md/semester_3/digital_logic/chapter-3

## Question

Simplify the Boolean function F(A,B,C,D) = Σm(0,1,2,4,7,8,9,10,12,15) and don't care condition (5,11,13) using K-Map and implement by only NOR gates.

## Short answer

Simplified POS: F = (B + C' + D')(B' + C' + D). NOR-only: invert B, C, D; use NOR gates to compute the sum terms, then NOR them together to get F. Requires six NOR gates.

## Answer

Given $F(A,B,C,D) = \Sigma m(0,1,2,4,7,8,9,10,12,15) + d(5,11,13)$.

#### K-map simplification (grouping 1s)

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

- Octet m(0,1,4,5,8,9,12,13) → $\overline{C}$
- Quad m(0,2,8,10) → $\overline{B}\,\overline{D}$
- Quad m(5,7,13,15) → $BD$

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

Variable A drops out completely.

#### POS form for NOR realization

NOR gates give a POS expression directly (NOR-NOR = OR-AND). The 0s are at cells 3, 6 and 14; grouping them (with don't-care 11):

- Pair of 0s M(3,11) → $(B+\overline{C}+\overline{D})$
- Pair of 0s M(6,14) → $(\overline{B}+\overline{C}+D)$

$$
F = (B+\overline{C}+\overline{D})(\overline{B}+\overline{C}+D)
$$

Check: the SOP and POS forms agree on every cell; e.g. cell 6 (B=1, C=1, D=0) makes the second sum term 0, so F = 0.

#### NOR-only circuit

$$
F = \overline{\overline{(B+\overline{C}+\overline{D})} + \overline{(\overline{B}+\overline{C}+D)}}
$$

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

B, C', D' ──[NOR]── P1  ─┐
                         ├──[NOR]── F
B', C', D ──[NOR]── P2  ─┘
```

Gates: 3 NOR inverters, 2 three-input NORs and 1 two-input NOR: 6 NOR gates. (Implementing the SOP form instead would need more gates.)

## Check yourself

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

1. SOP:
   A. C' + B'D' + BD
   B. C + BD
   C. A'+B'
   D. C' + A
   Answer: A. A drops out. (Key point: SOP)

2. Why POS for NOR?
   A. NOR is AND
   B. POS is smaller always
   C. NOR-NOR gives OR-AND form
   D. SOP impossible
   Answer: C. NOR-NOR equals OR-AND. (Key point: Why POS)

3. NOR gates used:
   A. 8
   B. 5
   C. 4
   D. 6
   Answer: D. 3 inverters, 2 three-input, 1 two-input. (Key point: Gate count)

## 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 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×
- [K-map Σ(0,1,4,8,10,11,12)+d, NAND-only](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Akmap-nand-only-01248) — Digital Logic, chapter 3
- [K-map Σ(0,2,4,5,11)+d, NOR-only](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Akmap-nor-only-02451) — 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

## 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