Boolean Logic
Logic gates, truth tables, logic expressions, circuits, interconversion and an interactive gate simulator.
1 Introduction to Boolean Logic
Boolean logic is a system of logic where every value is either TRUE (1) or FALSE (0). Logic gates are electronic components that take binary inputs and produce a binary output according to a specific rule.
In truth tables: 1 = TRUE and 0 = FALSE. These two notations are fully interchangeable — you will see both in Cambridge exams.
2 The Six Logic Gates
| Gate | Rule (plain English) | Output = 1 when… | |||
|---|---|---|---|---|---|
| NOT | Inverts the single input | AND | Output 1 only if BOTH inputs are 1 | OR | Output 1 if AT LEAST ONE input is 1 |
| NAND | Opposite of AND | NOR | Opposite of OR | XOR / EOR | Output 1 only if inputs are DIFFERENT |
3 Truth Tables
NOT Gate (1 input)
| A | NOT A |
|---|
AND Gate
| A | B | A AND B | |
|---|---|---|---|
| 0 | 0 | 1 | 1 |
OR Gate
| A | B | A OR B | |
|---|---|---|---|
| 0 | 0 | 1 | 1 |
NAND Gate
| A | B | A NAND B | |
|---|---|---|---|
| 0 | 0 | 1 | 1 |
NOR Gate
| A | B | A NOR B | |
|---|---|---|---|
| 0 | 0 | 1 | 1 |
XOR / EOR Gate
| A | B | A XOR B | |
|---|---|---|---|
| 0 | 0 | 1 | 1 |
Memory trick for XOR: "Exclusively different" — output is 1 only when the two inputs are NOT equal to each other.
4 Logic Expressions
A logic expression combines inputs using gate names to describe a circuit in text form. Example: X = (A AND B) OR (NOT C)
Evaluating an Expression — Worked Example
Evaluate X = (A AND B) OR (NOT C) for A=1, B=0, C=1:
- Evaluate innermost brackets first:
A AND B = 1 AND 0 = 0 - Evaluate NOT:
NOT C = NOT 1 = 0 - Combine with OR:
0 OR 0 = 0 - Therefore X = 0
5 Logic Circuits
A logic circuit is a diagram of interconnected gate symbols. For this syllabus, circuits have a maximum of three inputs and one output. No simplification is required.
Drawing a Circuit from an Expression
Expression: Z = (A OR B) AND (NOT A)
- Identify gates needed: 1× OR, 1× NOT, 1× AND
- Input A feeds into both the OR gate and the NOT gate
- Input B feeds into the OR gate
- Output of OR gate and output of NOT gate both feed into the AND gate
- AND gate output is labelled Z
Always label your inputs (A, B, C) and your final output clearly. Unlabelled inputs or outputs will lose marks in the exam.
6 Three-Input Circuits
With three inputs A, B, C there are 2³ = 8 rows in the truth table. Use this template — the bit pattern is systematic:
| A | B | C | Output | ||||
|---|---|---|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
Pattern: column A alternates every 4 rows, column B every 2 rows, column C every row. This guarantees all 8 combinations are covered systematically.
Worked Three-Input Example
Expression: W = (A AND B) OR (NOT C)
| A | B | C | A AND B | NOT C | W | ||
|---|---|---|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
7 Interconversion — the 3-Way Skill
The exam can ask you to convert between any combination of these four forms:
- Written formula using AND, OR, NOT, NAND, NOR, XOR
- e.g.
X = (A AND B) OR C - Use brackets to show order clearly
- Lists all possible input combinations
- 2 inputs → 4 rows; 3 inputs → 8 rows
- One output column filled for every row
- Diagram using standard gate symbols
- Inputs labelled A, B, C; output labelled
- Wires connect gate outputs to next gate inputs
- Written English description of the logic
- "Output is 1 when A is 1 and B is 0"
- Convert words → logic expression → circuit
Practise all directions: expression→circuit, circuit→truth table, truth table→expression, statement→all three. Any direction can appear in the exam — they carry equal weight.
8 Interactive Gate Simulator
Toggle inputs A and B to see all gate outputs update in real time.