Boolean Algebra Laws — TechAmbitionX

Cheat-sheet: fundamentals + derived laws.

Boolean algebra is the math of 0s and 1s. These rules let you simplify expressions and design efficient circuits.

Fundamental Laws

• Identity: A + 0 = A, A · 1 = A
• Null / Domination: A + 1 = 1, A · 0 = 0
• Idempotent: A + A = A, A · A = A
• Complement: A + A' = 1, A · A' = 0
• Involution (Double negation): (A')' = A

Algebraic Laws

• Commutative: A + B = B + A, A · B = B · A
• Associative: (A + B) + C = A + (B + C), (A · B) · C = A · (B · C)
• Distributive: A · (B + C) = A · B + A · C, A + (B · C) = (A + B) · (A + C)

Important Simplification Laws

• Absorption: A + (A · B) = A, A · (A + B) = A
• Complement / Complementarity: A + A' = 1, A · A' = 0
• De Morgan's Theorems: (A · B)' = A' + B', (A + B)' = A' · B'

Complement pair: A and A'

A	A'	A + A'	A · A'
0	1	1	0
1	0	1	0

De Morgan: (A · B)' = A' + B'

A	B	A · B	(A · B)'	A'	B'	A' + B'
0	0	0	1	1	1	1
0	1	0	1	1	0	1
1	0	0	1	0	1	1
1	1	1	0	0	0	0

Additional & Derived Laws (Bonus)

Consensus (Redundancy) Theorem

AB + A'C + BC = AB + A'C

The term BC is redundant given the other two — useful for removing unnecessary product terms.

Redundancy / Simplification Variant

A + A'B = A + B

This helps cut terms when a variable and its complement appear with other terms.

Transposition (Implication-like)

A + BC = (A + B)(A + C)

It's the distributive variant you use when converting between sum/product forms.

Duality Principle

Replace + with ·, · with +, 0 with 1 and 1 with 0 — any true identity stays true.

E.g., from A + 0 = A the dual is A · 1 = A. Duality is handy for deriving paired identities quickly.

Substitution Law

You can substitute equals for equals: if X = Y then replace X by Y everywhere to simplify.

Huntington Postulates (axiomatic basis)

A minimal set of axioms from which Boolean algebra can be built. Useful if you like math foundations, not required for typical DLD.

Useful Identities & Shortcuts

• DeMorgan + Distribution: Apply De Morgan to push complements inside or outside as needed.
• Look for complements first: Terms like A + A' or A · A' collapse immediately.
• Group terms: Use associative/commutative to rearrange before applying distributive/absorption.

Pro tip: When simplifying, scan for A + A', A · A', and then check for patterns of the Consensus theorem — they remove the most clutter.

Note: Many of the "extra" laws are derivable from the fundamentals. So if you memorize the core set and practice derivations, you can recreate others as well.

Disclaimer: Mistakes can occur, if you find one, Contact ASAP.
Verify with course material before exams.