---
title: "BCD-to-seven-segment decoder, segments a and e"
subject: "Digital Logic"
chapter: 4
key: "semester_3/digital_logic:bcd-seven-segment-a-e"
source: https://www.bctengineer.com/q/semester_3/digital_logic/bcd-seven-segment-a-e
site: BCT Engineering
---

# BCD-to-seven-segment decoder, segments a and e

Digital Logic · Chapter 4: Data Processing Circuits

- Asked in: 2079 Bhadra, 6 marks
- Video: [BCD TO SEVEN SEGMENT DECODER || DIGITAL ELECTRONICS](https://www.youtube.com/watch?v=U9dX2fdd7rc) — KTU ECE CLASSROOM
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/digital_logic/bcd-seven-segment-a-e
- Whole chapter: https://www.bctengineer.com/md/semester_3/digital_logic/chapter-4

## Question

Design the BCD to seven segment decoder. Obtain the simplest logic expressions for segments "a" and "e" also draw their circuits.

## Short answer

For BCD inputs A–D (A MSB), segment a = A + C + BD + B'D', segment e = D'(B'+C) = B'D' + CD'. Don't cares for 10–15 are used. Circuit for e uses one OR, one AND and an inverter on D.

## Answer

A **BCD-to-seven-segment decoder** takes a 4-bit BCD digit $ABCD$ ($A$ = MSB) and drives the seven segments a–g of a display so that the decimal digit 0–9 is shown. Inputs 1010–1111 never occur, so they are **don't cares**. (Common-cathode display: a 1 lights the segment.)

```text
   --a--
  |     |
  f     b
  |--g--|
  e     c
  |     |
   --d--
```

#### Truth table for segments a and e

| Digit | A B C D | a | e |
|---|---|---|---|
| 0 | 0000 | 1 | 1 |
| 1 | 0001 | 0 | 0 |
| 2 | 0010 | 1 | 1 |
| 3 | 0011 | 1 | 0 |
| 4 | 0100 | 0 | 0 |
| 5 | 0101 | 1 | 0 |
| 6 | 0110 | 1 | 1 |
| 7 | 0111 | 1 | 0 |
| 8 | 1000 | 1 | 1 |
| 9 | 1001 | 1 | 0 |

So $a = \Sigma m(0,2,3,5,6,7,8,9) + d(10\text{–}15)$ and $e = \Sigma m(0,2,6,8) + d(10\text{–}15)$.

#### K-map for a

```text
        CD=00 01  11  10
AB=00     1   0   1   1
AB=01     0   1   1   1
AB=11     X   X   X   X
AB=10     1   1   X   X
```

Groups: $A$ (rows 11, 10 with don't cares), $C$ (columns 11, 10), $BD$ (cells 5, 7, 13, 15), $B'D'$ (corners 0, 2, 8, 10).

$$
a = A + C + BD + B'D'
$$

#### K-map for e

```text
        CD=00 01  11  10
AB=00     1   0   0   1
AB=01     0   0   0   1
AB=11     X   X   X   X
AB=10     1   0   X   X
```

Groups: $B'D'$ (corners 0, 2, 8, 10) and $CD'$ (column 10: 2, 6, 14, 10).

$$
e = B'D' + CD' = D'(B' + C)
$$

#### Circuits

```text
a:  A --------------------+
    C --------------------+
    B --+[AND]------------+--[4-in OR]--> a
    D --+[   ]            |
    B'--+[AND]------------+
    D'--+[   ]

e:  B' --+--[OR]--+
    C  --+--[  ]  +--[AND]--> e
    D' -----------+--[   ]
```

$e$ needs only one OR gate, one AND gate and an inverter for $D$ ($B'$ is shared with segment a). Check: digit 6 (0110): $a = 0+1+0+0 = 1$, $e = D'(B'+C) = 1\cdot(0+1) = 1$, correct.

## Check yourself

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

1. What is done with inputs 1010-1111?
   A. Treated as 1
   B. Treated as 0
   C. Mapped to digit 8
   D. Treated as don't-cares
   Answer: D. They never occur in BCD. (Key point: Don't-cares)

2. What is the simplest expression for a?
   A. A + C + BD + B'D'
   B. A + B + C
   C. BD + B'D'
   D. A + C'D
   Answer: A. Groups A, C, BD and corners B'D' cover all minterms with don't-cares. (Key point: Segment a)

3. What is the simplest expression for e?
   A. A + C
   B. B'D' + CD'
   C. B'D' + BD
   D. CD + B'D'
   Answer: B. e = D'(B' + C), lit for 0, 2, 6, 8. (Key point: Segment e)

4. For which digits is segment e ON?
   A. 1, 3, 5, 7
   B. 0, 4, 8
   C. 0, 2, 6, 8
   D. 2, 3, 5, 9
   Answer: C. These are the minterms of e. (Key point: Digits for e)

## Related in Digital Logic

- [DEMUX vs decoder; seven-segment "b" segment](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Ademux-decoder-b-segment) — Digital Logic, chapter 4
- [Seven-segment decoder "b" segment](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Ab-segment-seven-segment) — Digital Logic, chapter 4, asked 2×
- [Seven-segment "b" and "d" segments](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Ab-d-segments) — Digital Logic, chapter 4
- [BCD-to-7-segment decoder; frequency counter](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Abcd-7segment-frequency-counter) — Digital Logic, chapter 11
- [Seven-segment "a" segment using NOR gates](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Aa-segment-nor-gates) — Digital Logic, chapter 4
- [Seven-segment "e" segment](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Ae-segment-seven-segment) — Digital Logic, chapter 4

## Related in other subjects

- [BCD counter 0-9999 on seven segment VHDL](https://www.bctengineer.com/question.md?key=semester_6%2Fembedded_system%3Avhdl-bcd-counter-seven-segment) — Embedded System, chapter 9
- [Seven segment display and clock counter](https://www.bctengineer.com/question.md?key=semester_6%2Fembedded_system%3Aseven-segment-clock-design) — Embedded System, chapter 9
- [Display 0 to 9 on seven segment](https://www.bctengineer.com/question.md?key=semester_6%2Fembedded_system%3A8051-seven-segment-0-9) — Embedded System, chapter 9, asked 2×
- [Display 99 to 00 on seven segment](https://www.bctengineer.com/question.md?key=semester_6%2Fembedded_system%3A8051-seven-segment-99-00) — Embedded System, chapter 9