---
title: "K-map Σ(0,2,4,5,11)+d, NOR-only"
subject: "Digital Logic"
chapter: 3
key: "semester_3/digital_logic:kmap-nor-only-02451"
source: https://www.bctengineer.com/q/semester_3/digital_logic/kmap-nor-only-02451
site: BCT Engineering
---

# K-map Σ(0,2,4,5,11)+d, NOR-only

Digital Logic · Chapter 3: Combinational Logic Circuits

- Asked in: 2081 Baisakh, 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-02451
- Whole chapter: https://www.bctengineer.com/md/semester_3/digital_logic/chapter-3

## Question

Implement the following function using K-Map. F(A,B,C,D) = Σ(0,2,4,5,11) + d(3,7,12,15). Implement the function using NOR gate only.

## Short answer

Simplified SOP: F = A'B'D' + A'B C' + CD. NOR-only: invert B, C, D; form P1 = NOR(A,B,D), P2 = NOR(A,B',C), P3 = NOR(C',D'); then F = NOR( NOR(P1,P2,P3) ), requiring eight NOR gates.

## Answer

Given $F(A,B,C,D) = \Sigma m(0,2,4,5,11) + d(3,7,12,15)$.

#### K-map simplification

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

Groups (don't-cares 3, 7, 15 are used for the quad; 12 is not needed):

- Pair m(0,2) → $\overline{A}\,\overline{B}\,\overline{D}$
- Pair m(4,5) → $\overline{A}B\overline{C}$
- Quad m(3,7,11,15) → $CD$

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

Check: 0, 2 by $\overline{A}\,\overline{B}\,\overline{D}$; 4, 5 by $\overline{A}B\overline{C}$; 11 by $CD$. (The minimal POS needs four sum terms, so SOP is simpler here.)

#### Realization 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.

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

where $P_1, P_2, P_3$ are the three first-level NOR outputs.

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

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

F' ──[NOR]── F   (inputs tied)
```

Gates used: 3 NOR inverters (for $\overline{B}, \overline{C}, \overline{D}$), three first-level NORs, one 3-input NOR and one NOR inverter: 8 NOR gates in all.

## Check yourself

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

1. Minimal SOP:
   A. A'B'D' + A'BC' + CD
   B. AB + CD
   C. A'D' + CD
   D. BC + A'
   Answer: A. Pairs and quad. (Key point: Simplified F)

2. CD quad uses:
   A. Minterms only
   B. Don't cares 3,7,15 with minterm 11
   C. Cell 12
   D. Cells 0,2
   Answer: B. m(3,7,11,15). (Key point: Quad)

3. SOP to NOR-only:
   A. Direct AND
   B. Single NOR
   C. First-level NORs, NOR to get F', then NOR inverter
   D. Use XOR
   Answer: C. Each product NOR of complemented literals. (Key point: NOR method)

4. Total NOR gates:
   A. 6
   B. 5
   C. 10
   D. 8
   Answer: D. 3 inverters, 3 NORs, 3-input NOR, inverter. (Key point: Gate count)

## Related in Digital Logic

- [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 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 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 Π(0,1,4,7,8,10,11,12)+d, NAND](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Akmap-maxterm-nand-0147) — 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