---
title: "Verify Boolean identities"
subject: "Basic Electronics Engineering"
chapter: 6
key: "semester_2/basic_electronics_engineering:verify-boolean-identities"
source: https://www.bctengineer.com/q/semester_2/basic_electronics_engineering/verify-boolean-identities
site: BCT Engineering
---

# Verify Boolean identities

Basic Electronics Engineering · Chapter 6: Digital Electronics

- Asked in: 2069 Bhadra, 2+2 marks
- Video: [Boolean Algebra 1 – The Laws of Boolean Algebra](https://www.youtube.com/watch?v=EPJf4owqwdA) — Computer Science Lessons
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/basic_electronics_engineering/verify-boolean-identities
- Whole chapter: https://www.bctengineer.com/md/semester_2/basic_electronics_engineering/chapter-6

## Question

Verify the following: i) AB + A'C = (A+C)(A'+B)  ii) XY + X'Z + YZ = XY + X'Z

## Short answer

i) Expanding (A+C)(A'+B) gives AB + A'C, so AB + A'C = (A+C)(A'+B). ii) XY + X'Z + YZ = XY + X'Z by consensus; multiplying YZ by (X+X') and simplifying shows YZ is redundant.

## Answer

#### i) AB + A'C = (A + C)(A' + B)

Take the right-hand side (RHS):

```text
(A + C)(A' + B) = AA' + AB + CA' + CB
                = 0 + AB + A'C + BC
                = AB + A'C + BC(A + A')
                = AB + A'C + ABC + A'BC
                = AB(1 + C) + A'C(1 + B)
                = AB + A'C
```

RHS = LHS. Verified.

#### ii) XY + X'Z + YZ = XY + X'Z (consensus theorem)

Take the left-hand side (LHS) and multiply YZ by (X + X') = 1:

```text
XY + X'Z + YZ = XY + X'Z + YZ(X + X')
              = XY + X'Z + XYZ + X'YZ
              = XY(1 + Z) + X'Z(1 + Y)
              = XY + X'Z
```

LHS = RHS. Verified.

## Check yourself

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

1. To verify (A + C)(A' + B) = AB + A'C, the answer expands:
   A. The right side and simplifies to AB + A'C
   B. The left side using De Morgan
   C. Both sides by K-map only
   D. The left side by adding all minterms
   Answer: A. The answer states this: The right side and simplifies to AB + A'C. (Key point: Part i method)

2. In part i, AA' equals:
   A. 1
   B. 0
   C. A
   D. A'
   Answer: B. The answer states this: 0. (Key point: Useful identity)

3. XY + X'Z + YZ = XY + X'Z shows that:
   A. XY is redundant
   B. X'Z is redundant
   C. YZ is redundant
   D. All terms are needed
   Answer: C. The answer states this: YZ is redundant. (Key point: Consensus theorem)

4. In part ii, multiplying YZ by (X + X') is allowed because:
   A. X + X' = 0
   B. X . X' = 1
   C. X + X' = X
   D. X + X' = 1
   Answer: D. The answer states this: X + X' = 1. (Key point: Multiplying by X + X')

## Related in Basic Electronics Engineering

- [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

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