Boolean Algebra Theorems – Simplifying Logic Made Easy!

Boolean algebra uses a set of rules and theorems to manipulate logical expressions. These rules are the foundation of digital circuit design and simplification.


1. Identity Theorems

  • A + 0 = A

  • A · 1 = A

These mean that OR-ing with 0 or AND-ing with 1 doesn’t change the value.

2. Null Theorems

  • A + 1 = 1

  • A · 0 = 0

OR-ing anything with 1 always gives 1; AND-ing with 0 gives 0.

3. Idempotent Theorems

  • A + A = A

  • A · A = A

Repeated use of the same variable doesn’t change the result.

4. Complement Theorems

  • A + A’ = 1

  • A · A’ = 0

A variable OR-ed with its complement is always 1; AND-ed is always 0.

5. Involution Law

  • (A’)’ = A

Double negation brings back the original value.

6. Commutative Laws

  • A + B = B + A

  • A · B = B · A

Order doesn’t matter for OR and AND operations.

7. Associative Laws

  • A + (B + C) = (A + B) + C

  • A · (B · C) = (A · B) · C

Grouping doesn’t affect the result of OR or AND.

8. Distributive Laws

  • A · (B + C) = A·B + A·C

  • A + (B · C) = (A + B) · (A + C)

These help in expanding or factoring expressions.

9. Absorption Theorems

  • A + A·B = A

  • A · (A + B) = A

These remove redundant terms to simplify expressions.

10. DeMorgan’s Theorems

  • (A · B)’ = A’ + B’

  • (A + B)’ = A’ · B’

Important for converting between AND/OR and NAND/NOR.

Practical Use

These theorems are widely used in:

  • Simplifying logic circuits

  • Reducing gate count

  • Designing combinational logic (e.g., multiplexers, decoders)

  • Optimizing software logic (especially in embedded systems)

Stay tuned to hobitronics.blog

To know more about Logic Gates: Understanding Logic Gates

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