---
title: "Σm(0,1,3,4,8,9,15)+d using 8:1 MUX"
subject: "Digital Logic"
chapter: 4
key: "semester_3/digital_logic:mux-8to1-function-dont-care"
source: https://www.bctengineer.com/q/semester_3/digital_logic/mux-8to1-function-dont-care
site: BCT Engineering
---

# Σm(0,1,3,4,8,9,15)+d using 8:1 MUX

Digital Logic · Chapter 4: Data Processing Circuits

- Asked in: 2082 Baisakh, 4 marks
- Video: [Implementation of Boolean Function using Multiplexers || 8:1 || 4:1 || implementing boolean function](https://www.youtube.com/watch?v=AwZCNhxgKFY) — Sudhakar Atchala
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/digital_logic/mux-8to1-function-dont-care
- Whole chapter: https://www.bctengineer.com/md/semester_3/digital_logic/chapter-4

## Question

Implement the following function with 8:1 multiplexer F(A,B,C,D) = Σm(0,1,3,4,8,9,15) + dΣ(2,6,13).

## Short answer

Connect A,B,C as selects. Set I0=I1=I4=1, I3=I5=I6=0, I2=D', I7=D. Don’t-cares at 2,6,13 are chosen to make inputs constant or simple functions of D, realizing F(A,B,C,D) = Σm(0,1,3,4,8,9,15)+d(2,6,13).

## Answer

Use $A, B, C$ as the select lines $S_2S_1S_0$ and $D$ (or $D'$, 0, 1) on the data inputs. Input $I_j$ covers minterms $2j$ ($D=0$) and $2j+1$ ($D=1$).

- Minterms: 0, 1, 3, 4, 8, 9, 15
- Don't-cares: 2, 6, 13 (each can be taken as 0 or 1, whichever gives a simpler input)

#### Implementation table (X = don't care)

| | $I_0$ | $I_1$ | $I_2$ | $I_3$ | $I_4$ | $I_5$ | $I_6$ | $I_7$ |
|---|---|---|---|---|---|---|---|---|
| $D'$ | **0** | X(2) | **4** | X(6) | **8** | 10 | 12 | 14 |
| $D$ | **1** | **3** | 5 | 7 | **9** | 11 | X(13) | **15** |
| Input | 1 | 1 | $D'$ | 0 | 1 | 0 | 0 | $D$ |

Choices for the don't-cares:

- $I_1$: minterm 3 is 1; taking d(2) = 1 gives $I_1 = 1$ (no gate needed).
- $I_3$: minterm 7 is 0; taking d(6) = 0 gives $I_3 = 0$.
- $I_6$: minterm 12 is 0; taking d(13) = 0 gives $I_6 = 0$.

#### Circuit

```text
             +--------+
  1 -------->| I0     |
  1 -------->| I1     |
  D --[NOT]->| I2     |
  0 -------->| I3 8:1 |----> F
  1 -------->| I4 MUX |
  0 -------->| I5     |
  0 -------->| I6     |
  D -------->| I7     |
             +--------+
              A  B  C
```

**Check:** $ABCD = 0100$ (4): select 010 → $I_2 = D' = 1$, $F = 1$. $ABCD = 0101$ (5): $I_2 = D' = 0$, $F = 0$. $ABCD = 1111$ (15): $I_7 = D = 1$. All required minterms give 1 and all other non-don't-care minterms give 0.

## Check yourself

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

1. Which variables are used as select lines for F(A,B,C,D)?
   A. D, C, B as selects and A as data
   B. A, B, C as selects and D as data
   C. A, B as selects and C, D as data
   D. All four as selects
   Answer: B. Each data input Ij covers minterms 2j and 2j+1, which differ in D. (Key point: Select lines)

2. How are the don't-cares 2, 6 and 13 used?
   A. They are all taken as 1
   B. They are all taken as 0
   C. Each is taken as 0 or 1, whichever gives the simplest data input
   D. They are ignored and the output is left floating
   Answer: C. Choosing d(2)=1 makes I1=1, and d(6)=0, d(13)=0 make I3=0 and I6=0. (Key point: Role of don't-cares)

3. What is applied to I2 (minterms 4 and 5)?
   A. D
   B. 1
   C. 0
   D. D'
   Answer: D. Minterm 4 (D=0) is in F and 5 (D=1) is not, so I2 = D' (through a NOT gate). (Key point: Data input I2)

4. What is applied to I7 (minterms 14 and 15)?
   A. D
   B. D'
   C. 1
   D. 0
   Answer: A. Minterm 14 is not in F and 15 is, so the output equals D. (Key point: Data input I7)

## Related in Digital Logic

- [32:1 MUX tree; implement Σm(0,1,3,8,9,13,15)](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Amux-32to1-tree-function-15) — Digital Logic, chapter 4
- [32:1 MUX tree from 4:1; implement Σ(0,1,3,8,9,13)](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Amux-32to1-tree-function) — Digital Logic, chapter 4
- [Σ(0,2,3,6,7,8,12,13,15) using 8×1 MUX](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Amux-8x1-function-0236) — Digital Logic, chapter 4
- [K-map ΠM(0,2,4,6,7,8,10,13,15) and 8:1 MUX](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Akmap-pom-8to1-mux) — Digital Logic, chapter 3
- [Function Σm(0,2,3,5,7,8,10,13,15) using 1:8 DEMUX](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Ademux-1to8-realize-function) — Digital Logic, chapter 4
- [Multiplexer tree; 16:1 MUX from 4:1](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Amux-16to1-from-4to1) — Digital Logic, chapter 4

## Related in other subjects

- [Multiplexer and 4:1 MUX](https://www.bctengineer.com/question.md?key=semester_2%2Fbasic_electronics_engineering%3Amultiplexer-4-1) — Basic Electronics Engineering, chapter 6, asked 2×