Digital Logic Foundations
Gates and Boolean Algebra
Logic gates are the fundamental building blocks of digital circuits. Boolean Algebra provides the mathematical framework to analyse and design gate networks.
What is a Gate?
A gate is an electronic component (transistors connected together) whose output is determined by its inputs. A single chip may contain anywhere from 4 gates to over 10 million.
Basic Logic Gates
NOT Gate
Inverts the input. Also written with prime: c = a'
| a | c |
|---|---|
| 0 | 1 |
| 1 | 0 |
OR Gate
Output is 1 if either input is 1. Operator '+' preferred over '∨'.
| a | b | c |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 1 |
AND Gate
Output is 1 only when both inputs are 1. Operator '.' preferred over '∧'.
| a | b | c |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
XOR Gate
Exclusive OR — output is 1 when inputs differ. Basis for modulo-2 addition.
| a | b | c |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |
Inverted (N) Variants
NAND
NOT of AND. Universal gate — can build any other gate from NAND alone.
| a | b | c |
|---|---|---|
| 0 | 0 | 1 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |
NOR
NOT of OR. Also a universal gate.
XNOR
Output is 1 when both inputs are equal.
Boolean Arithmetic Rules
Combining Gates
Logic gates can be chained to build any desired function. Example: the circuit for A·B + C·D uses two AND gates feeding into an OR gate. The output depends only on current inputs — there is no memory of past state.
Gate Symbols in Logisim
| Symbol | Operator | Logisim icon | Description |
|---|---|---|---|
| · | AND | Output high only when both inputs are high | |
| + | OR | Output high if either input is high | |
| ¬ | NOT | Inverts its input | |
| ⊕ | XOR | Output high when exactly one input is high | |
| ↑ | NAND | Inverted AND output | |
| ↓ | NOR | Inverted OR output | |
| ⊙ | XNOR | Inverted XOR output |