Boolean Algebra and Logic Simplification
Boolean algebra
Boolean algebra manipulates logic expressions to simplify circuits. Simpler expressions mean fewer logic gates, which makes hardware cheaper, faster and more reliable. Notation: A·B = AND, A + B = OR, Ā (overbar) = NOT.
The logic laws
| Law | Rule |
|---|---|
| Identity | A·1 = A, A + 0 = A |
| Null | A·0 = 0, A + 1 = 1 |
| Idempotent | A·A = A, A + A = A |
| Inverse | A·Ā = 0, A + Ā = 1 |
| Commutative | A·B = B·A, A + B = B + A |
| Associative | A·(B·C) = (A·B)·C |
| Distributive | A·(B + C) = A·B + A·C |
| Absorption | A + A·B = A, A·(A + B) = A |
De Morgan's laws
The two most-tested rules for simplifying negated expressions:
NOT(A AND B) = NOT A OR NOT B → (A·B)‾ = Ā + B̄
NOT(A OR B) = NOT A AND NOT B → (A + B)‾ = Ā·B̄Method: break (or join) the bar, and swap AND ↔ OR. Useful for rearranging circuits to use fewer, or a single type of, gate.
Karnaugh maps (K-maps)
A Karnaugh map is a grid method for minimising a Boolean expression visually:
1. Draw a grid with inputs arranged in Gray code order (only one bit changes between adjacent cells).
2. Fill in a 1 for each input combination that gives an output of 1.
3. Group the 1s into the largest possible rectangular blocks of size 1, 2, 4, 8… (groups may wrap around edges and overlap).
4. Each group becomes a simplified term (the variables that stay constant across the group); OR the terms together.
K-maps make it easy to spot the simplest expression without long algebra.
Worked example
Simplify A + A·B using the absorption law, then check with a truth table idea.
- Absorption: A + A·B = A. (Whenever A is 1 the whole thing is 1; whenever A is 0 it's A·B = 0 = A, so the result always equals A.) ✓
Common mistakes
- Applying De Morgan's without swapping the operator (AND↔OR).
- In K-maps, making groups that aren't powers of two, or missing larger groups that give more simplification.
- Confusing the OR symbol + with arithmetic addition.
Exam tips
- Learn De Morgan's laws and the key simplification laws (absorption, distributive, identity).
- For K-maps, always group the largest blocks of 1s (and remember wrap-around).
- Show each simplification step and name the law used for method marks.
Key facts to remember
- Boolean algebra simplifies expressions → fewer gates; notation · = AND, + = OR, overbar = NOT.
- De Morgan's: (A·B)‾ = Ā + B̄ and (A+B)‾ = Ā·B̄ — break the bar and swap the operator.
- Karnaugh maps minimise expressions by grouping 1s into the largest power-of-two blocks (Gray code order).