---
title: "K-map Σm(7,9,12,13,14,15)+d, NOR-only"
subject: "Digital Logic"
chapter: 3
key: "semester_3/digital_logic:kmap-nor-sum-7-9-12"
source: https://www.bctengineer.com/q/semester_3/digital_logic/kmap-nor-sum-7-9-12
site: BCT Engineering
---

# K-map Σm(7,9,12,13,14,15)+d, NOR-only

Digital Logic · Chapter 3: Combinational Logic Circuits

- Asked in: 2079 Bhadra, 6 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-sum-7-9-12
- Whole chapter: https://www.bctengineer.com/md/semester_3/digital_logic/chapter-3

## Question

Simplify the following Boolean function using K-map and draw the circuit of simplified expression using NOR gates only. F = Σm(7,9,12,13,14,15) + don't care (0,2,3,5).

## Short answer

F = AB + BD + A C' D. For NOR-only: invert A,B,D; P1 = NOR(A',B'), P2 = NOR(B',D'), P3 = NOR(A',C,D'); F = NOR( NOR(P1,P2,P3) ) via a final NOR inverter. Total eight NOR gates.

## Answer

Given $F(A,B,C,D) = \Sigma m(7,9,12,13,14,15) + d(0,2,3,5)$.

#### K-map simplification

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

Groups (don't-care 5 is used in the BD quad; 0, 2, 3 are not needed):

- Quad m(12,13,14,15) → $AB$
- Pair m(9,13) → $A\overline{C}D$
- Quad m(5,7,13,15) → $BD$

$$
F = AB + BD + A\overline{C}D
$$

Check: 12, 13, 14, 15 by AB; 7 (and 5, 13, 15) by BD; 9 by $A\overline{C}D$.

#### Circuit 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.

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

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

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

F' ──[NOR]── F   (inputs tied)
```

Gates: 3 NOR inverters ($\overline{A}, \overline{B}, \overline{D}$), 3 first-level NORs, a 3-input NOR giving $\overline{F}$ and a final NOR inverter.

## Check yourself

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

1. Simplified F:
   A. AB + BD + AC'D
   B. AB + CD
   C. BD + A'
   D. AB + AD
   Answer: A. Three groups. (Key point: Simplified F)

2. AB quad:
   A. m(4,5,6,7)
   B. m(12,13,14,15)
   C. m(8..11)
   D. m(0..3)
   Answer: B. A=B=1. (Key point: AB quad)

3. BD used don't care:
   A. 0
   B. 2
   C. 5
   D. 3
   Answer: C. m(5,7,13,15). (Key point: BD quad)

4. NOR-only inverts product to NOR of:
   A. true literals
   B. XOR
   C. nothing
   D. complemented literals
   Answer: D. AB=(A'+B')'. (Key point: NOR)

## 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
- [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
- [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 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(0,1,2,4,6,8,9,12,13,14)](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Aminterm-maxterm-kmap-01246) — 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