Boolean Logic
Boolean logic is the mathematical foundation of all digital circuits. Every decision a computer makes — from adding numbers to rendering games — is built from combinations of simple logic gates. At Grade 12 level, you need to be able to build logic circuits from a problem description, construct truth tables, and derive Boolean expressions.
Learning Objectives
- 12.3.3.2 Build logical structures
- 12.3.3.4 Analyze logical structures
Boolean Values
Boolean logic uses only two values:
| True | False |
|---|---|
| 1 | 0 |
| ON | OFF |
| HIGH | LOW |
| YES | NO |
Basic Logic Gates
1 AND Gate
Expression: X = A AND B (also written A · B or A ∧ B)
Rule: Output is 1 only if BOTH inputs are 1.
| A | B | X = A AND B |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
Real-world analogy: Two switches in series — both must be ON for the light to work.
2 OR Gate
Expression: X = A OR B (also written A + B or A ∨ B)
Rule: Output is 1 if at least one input is 1.
| A | B | X = A OR B |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 1 |
Real-world analogy: Two switches in parallel — either one can turn on the light.
3 NOT Gate (Inverter)
Expression: X = NOT A (also written A̅ or ¬A)
Rule: Output is the opposite of the input.
| A | X = NOT A |
|---|---|
| 0 | 1 |
| 1 | 0 |
Derived Gates
4 NAND Gate
Expression: X = NOT (A AND B) (opposite of AND)
Rule: Output is 0 only if BOTH inputs are 1.
| A | B | X = A NAND B |
|---|---|---|
| 0 | 0 | 1 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |
Why NAND is important: NAND is a universal gate — any logic circuit can be built using only NAND gates.
5 NOR Gate
Expression: X = NOT (A OR B) (opposite of OR)
| A | B | X = A NOR B |
|---|---|---|
| 0 | 0 | 1 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 0 |
NOR is also a universal gate.
6 XOR Gate (Exclusive OR)
Expression: X = A XOR B (also written A ⊕ B)
Rule: Output is 1 if inputs are different.
| A | B | X = A XOR B |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |
XOR = "one or the other, but not both."
Gate Summary Reference
| Gate | Expression | Output = 1 when… | Inputs |
|---|---|---|---|
| AND | A · B | Both A and B are 1 | 2+ |
| OR | A + B | At least one is 1 | 2+ |
| NOT | A̅ | A is 0 | 1 |
| NAND | (A · B)̅ | Not both are 1 | 2+ |
| NOR | (A + B)̅ | Both are 0 | 2+ |
| XOR | A ⊕ B | Inputs are different | 2 |
Building & Analyzing Circuits
✓ Worked Example — Derive the Truth Table from an Expression
Given: X = (A AND B) OR (NOT C)
Step 1: List all input combinations (2³ = 8 rows for 3 inputs)
Step 2: Calculate intermediate values
Step 3: Combine to get final output X
A | B | C | A AND B | NOT C | X = (A AND B) OR (NOT C)
----+-----+-----+---------+-------+-------------------------
0 | 0 | 0 | 0 | 1 | 1
0 | 0 | 1 | 0 | 0 | 0
0 | 1 | 0 | 0 | 1 | 1
0 | 1 | 1 | 0 | 0 | 0
1 | 0 | 0 | 0 | 1 | 1
1 | 0 | 1 | 0 | 0 | 0
1 | 1 | 0 | 1 | 1 | 1
1 | 1 | 1 | 1 | 0 | 1✓ Worked Example — Build a Circuit from a Problem
Problem: An alarm (X) should sound when: the door is open (A=1) AND the system is armed (B=1), OR the panic button is pressed (C=1).
Step 1: Identify the conditions:
Condition 1: A AND B (door open AND armed)
Condition 2: C (panic button)
Combined: (A AND B) OR C
Step 2: Write the Boolean expression:
X = (A · B) + C
Step 3: Build the circuit:
A ──┐
├── AND ──┐
B ──┘ ├── OR ── X
C ────────────┘
Step 4: Truth table (verify):
A B C | A·B | X=(A·B)+C
0 0 0 | 0 | 0
0 0 1 | 0 | 1
0 1 0 | 0 | 0
0 1 1 | 0 | 1
1 0 0 | 0 | 0
1 0 1 | 0 | 1
1 1 0 | 1 | 1 ← door open AND armed
1 1 1 | 1 | 1 ← both conditionsPitfalls & Common Errors
Confusing OR with XOR
OR gives 1 when one or both inputs are 1. XOR gives 1 when exactly one input is 1. When both are 1: OR=1, XOR=0.
Wrong Number of Rows
For n inputs, you need 2n rows. 2 inputs = 4 rows, 3 inputs = 8 rows, 4 inputs = 16 rows. Missing rows = lost marks.
Operator Precedence
Boolean precedence: NOT first, then AND, then OR. So A + B · C̅ means
A + (B · (NOT C)), not (A + B) · (NOT C).
Pro-Tips for Exams
Logic Circuit Strategy
- Always show intermediate columns in truth tables — examiners award method marks
- When analyzing a circuit: work from inputs to outputs, gate by gate
- When building a circuit: start from the output and work backwards to inputs
- Double-check your truth table by counting: an AND gate mostly produces 0s, an OR gate mostly produces 1s
- Remember precedence: NOT → AND → OR
Graded Tasks
Draw the truth tables for AND, OR, NOT, NAND, NOR, and XOR gates.
Build a truth table for: X = (A OR B) AND (NOT C)
A vending machine dispenses a drink when: (coin is inserted AND correct button is pressed) OR the override switch is on. Write the Boolean expression and draw the truth table.
Given the expression X = NOT(A AND B) OR C, prove using a truth table that this is NOT the same as X = NOT A AND NOT B OR C (i.e., De Morgan's law changes the expression).
Design a security system with 3 inputs: motion sensor (M), door sensor (D), and armed switch (S). The alarm sounds when: the system is armed AND either the motion sensor or door sensor is triggered. Write the expression, draw the circuit, and complete the truth table.