Boolean Algebra and Logic Gate Circuits

At A-Level you must go well beyond drawing simple gate circuits. You need to be able to write and simplify Boolean expressions, apply Boolean algebra laws, use Karnaugh maps (K-maps) to minimise expressions, and convert between truth tables, Boolean expressions, and gate circuits.

Boolean Operators and Truth Tables

The core Boolean operations are:

Operator	Symbol	Logic Gate	Definition
AND	A . B (or A ∧ B)	AND gate	Output is 1 only if both inputs are 1
OR	A + B (or A ∨ B)	OR gate	Output is 1 if at least one input is 1
NOT	A̅ (or ¬A)	NOT gate (inverter)	Output is the complement of the input
NAND	(A . B)̅	NAND gate	NOT AND — output is 0 only if both inputs are 1
NOR	(A + B)̅	NOR gate	NOT OR — output is 0 if at least one input is 1
XOR	A ⊕ B	XOR gate	Output is 1 if inputs are different

Truth Tables

A	B	A . B	A + B	A̅	A ⊕ B	(A . B)̅	(A + B)̅
0	0	0	0	1	0	1	1
0	1	0	1	1	1	1	0
1	0	0	1	0	1	1	0
1	1	1	1	0	0	0	0

Boolean Algebra Laws

You must know and be able to apply these laws to simplify expressions:

Basic Laws

Law	AND form	OR form
Identity	A . 1 = A	A + 0 = A
Null (Annulment)	A . 0 = 0	A + 1 = 1
Idempotent	A . A = A	A + A = A
Complement	A . A̅ = 0	A + A̅ = 1
Double negation	(A̅)̅ = A

Algebraic Laws

Law	AND form	OR form
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)

De Morgan's Laws

These are essential for simplification:

(A . B)̅ = A̅ + B̅ — the complement of AND is OR with complemented inputs.
(A + B)̅ = A̅ . B̅ — the complement of OR is AND with complemented inputs.

Exam Tip: De Morgan's Laws are tested almost every year. To apply them: (1) break the bar, (2) change the operator (AND ↔ OR), (3) complement each term. Practise applying them to multi-variable expressions.

Simplifying Boolean Expressions — Algebraic Method

Example: Simplify Q = A.B + A.B̅

Factor out A: Q = A . (B + B̅)
Apply complement law (B + B̅ = 1): Q = A . 1
Apply identity law (A . 1 = A): Q = A

Boolean Algebra and Logic Gate Circuits (A-Level depth, Karnaugh maps)