Boolean Functions
A Boolean function maps each input combination to an output of 0 or 1.
Description
A rule assigning a 0 or 1 output to every combination of input variables. It is the precise specification of what a combinational circuit must compute. Capture it as a truth table, an algebraic expression, or a gate network — interchangeably.
- Expression: e.g. F = AB + C.
- Truth table: output listed for all 2ⁿ input rows.
- Circuit: gates wired to realize the expression.
- F′ is found by applying DeMorgan throughout, or by swapping 0s/1s in the truth table.
- What: A rule assigning a 0 or 1 output to every combination of input variables.
- Why: It is the precise specification of what a combinational circuit must compute.
- How: Capture it as a truth table, an algebraic expression, or a gate network — interchangeably.
- Where: Every combinational design starts as a Boolean function.
- When: At specification time, before choosing an implementation.
- Analogy — A Boolean function is like a recipe's required outcome ('dish must be salty when X'); the expression and the circuit are two different kitchens that produce the same dish.
At a glance
What
A rule assigning a 0 or 1 output to every combination of input variables.
Why
It is the precise specification of what a combinational circuit must compute.
How
Capture it as a truth table, an algebraic expression, or a gate network — interchangeably.
Where
Every combinational design starts as a Boolean function.
When
At specification time, before choosing an implementation.
Think of it like…
A Boolean function is like a recipe's required outcome ('dish must be salty when X'); the expression and the circuit are two different kitchens that produce the same dish.
Three equivalent views
- Expression: e.g. F = AB + C.
- Truth table: output listed for all 2ⁿ input rows.
- Circuit: gates wired to realize the expression.
Complement of a function
- F′ is found by applying DeMorgan throughout, or by swapping 0s/1s in the truth table.
Example: F = AB + C
| A | B | C | F |
|---|---|---|---|
| 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 1 | 0 | 0 |
| 0 | 1 | 1 | 1 |
| 1 | 0 | 0 | 0 |
| 1 | 0 | 1 | 1 |
| 1 | 1 | 0 | 1 |
| 1 | 1 | 1 | 1 |
Explore single-gate functions
▶ live simulatorClick a terminal (A/B) to toggle it · glowing wires carry a logic 1 · the lamp is output Y
| A | B | Y |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
The 5 Whys
- 1
Why formalize a function? To specify a circuit unambiguously.
- 2
Why multiple representations? Each view suits a different design step.
- 3
Why is the truth table canonical? It lists behavior for every input — no ambiguity.
- 4
Why simplify the expression? Smaller expressions → smaller circuits.
- 5
Root cause: a function is the contract; the circuit is one of many valid implementations.
Cheat sheet
Working principle
- Capture it as a truth table, an algebraic expression, or a gate network — interchangeably.
- A rule assigning a 0 or 1 output to every combination of input variables.
Formulas & Boolean expressions
- Expression: e.g. F = AB + C.
- Truth table: output listed for all 2ⁿ input rows.
- F′ is found by applying DeMorgan throughout, or by swapping 0s/1s in the truth table.
Key facts
- Expression: e.g. F = AB + C.
- F′ is found by applying DeMorgan throughout, or by swapping 0s/1s in the truth table.
Why it exists
- Root cause: a function is the contract; the circuit is one of many valid implementations.