---
title: "K-map: 3-variable five-term expression"
subject: "Basic Electronics Engineering"
chapter: 6
key: "semester_2/basic_electronics_engineering:kmap-xyz-five-term"
source: https://www.bctengineer.com/q/semester_2/basic_electronics_engineering/kmap-xyz-five-term
site: BCT Engineering
---

# K-map: 3-variable five-term expression

Basic Electronics Engineering · Chapter 6: Digital Electronics

- Asked in: 2070 Bhadra, 4 marks
- Video: [Lec -12: What is K-Map | Design K-Map | 3 Variable K-Map](https://www.youtube.com/watch?v=36LjWW-BSyU) — Gate Smashers
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/basic_electronics_engineering/kmap-xyz-five-term
- Whole chapter: https://www.bctengineer.com/md/semester_2/basic_electronics_engineering/chapter-6

## Question

Simplify the expression using k-map F(x,y,z) = xyz + x'y'z + xy'z' + x'y'z' + x'yz

## Short answer

F = Σ(0,1,3,4,7) groups: pair (m0,m1) yields x'y', (m0,m4) yields y'z', (m3,m7) yields yz. Simplified: F = x'y' + y'z' + yz.

## Answer

F = xyz + x'y'z + xy'z' + x'y'z' + x'yz

Minterms: xyz = m7, x'y'z = m1, xy'z' = m4, x'y'z' = m0, x'yz = m3.
F = Σm(0, 1, 3, 4, 7).

#### K-map

```text
          yz
        00  01  11  10
 x = 0 | 1 | 1 | 1 | 0 |
 x = 1 | 1 | 0 | 1 | 0 |
```

#### Grouping

1. Pair (m0, m1): x = 0, y = 0 gives **x'y'**.
2. Pair (m0, m4): y = 0, z = 0 gives **y'z'** (the two cells of the 00 column).
3. Pair (m3, m7): y = 1, z = 1 gives **yz**.

All five 1s are covered.

**Answer: F = x'y' + y'z' + yz**

## Check yourself

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

1. xyz, x'y'z, xy'z', x'y'z', x'yz correspond to:
   A. m7, m1, m4, m0, m3
   B. m7, m1, m5, m0, m3
   C. m6, m1, m4, m0, m3
   D. m7, m2, m4, m0, m3
   Answer: A. The answer states this: m7, m1, m4, m0, m3. (Key point: Minterms)

2. The pair (m0, m1) gives:
   A. y'z'
   B. x'y'
   C. x'z
   D. yz
   Answer: B. The answer states this: x'y'. (Key point: Pair (m0, m1))

3. The pair (m0, m4) gives:
   A. x'y'
   B. xz'
   C. y'z'
   D. y'z
   Answer: C. The answer states this: y'z'. (Key point: Pair (m0, m4))

4. The simplified function is:
   A. F = x'y' + yz
   B. F = x'y' + y'z'
   C. F = y'z' + yz
   D. F = x'y' + y'z' + yz
   Answer: D. The answer states this: F = x'y' + y'z' + yz. (Key point: Result)

## Related in Basic Electronics Engineering

- [K-map: x'yz + x'y'z + xyz with gates](https://www.bctengineer.com/question.md?key=semester_2%2Fbasic_electronics_engineering%3Akmap-xyz-realize-gates) — Basic Electronics Engineering, chapter 6, asked 2×
- [K-map and universal gate: xy + x'z + yz](https://www.bctengineer.com/question.md?key=semester_2%2Fbasic_electronics_engineering%3Akmap-universal-gate-xy) — Basic Electronics Engineering, chapter 6
- [K-map: two 3-variable functions](https://www.bctengineer.com/question.md?key=semester_2%2Fbasic_electronics_engineering%3Akmap-two-expressions) — Basic Electronics Engineering, chapter 6
- [K-map: 4-variable minterm list](https://www.bctengineer.com/question.md?key=semester_2%2Fbasic_electronics_engineering%3Akmap-4var-sigma) — Basic Electronics Engineering, chapter 6

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