A Boolean function assigns an output, 0 or 1, to every combination of its inputs. A function of A, B and C is often written F(A, B, C).
The function is the behavior, not the formula. Its truth table pins it down completely. Many different expressions can describe the same function: , and A are three expressions for one function of A and B.
That distinction drives the whole course:
- A combinational circuit computes a Boolean function.
- Designing a circuit means choosing one expression for the function.
- Simplifying means finding a cheaper expression for the same function.
With n inputs there are 2ⁿ rows, and each row's output can be 0 or 1 independently, so there are 2 to the power 2ⁿ different functions: 16 functions of 2 variables, and 256 of 3 variables.
Every function can be written in a standard way, as a sum of minterms or a product of maxterms.
| A | ||||
|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 1 | 0 | 1 | 1 | 1 |
| 1 | 1 | 1 | 1 | 1 |