Boolean Algebra
Boolean algebra is a way of expressing operations on true / false booleans with a mathematical syntax. It can be simplified just like any other kind of mathematics, and this simplifies the underlying circuit, thus reducing how many components the circuit requires. This causes the ciruit to be:
- Cheaper
- More efficient
- With less gates: more reliable.
Truth tables
Truth tables are tables used to model each possible value of a boolean expression to calculate its possible values.
Operations
NOT
| A | NOT A |
|---|---|
| 0 | 1 |
| 1 | 0 |
OR
| A | B | A OR B |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 1 |
NOR
| A | B | A OR B |
|---|---|---|
| 0 | 0 | 1 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 0 |
AND
| A | B | A AND B |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
NAND
| A | B | A NAND B |
|---|---|---|
| 0 | 0 | 1 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |
XOR
| A | B | A XOR B |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |
Laws
The Identity law
These are self-explanatory key laws.
The Absorption Law
The Null Law
The Commutative law
This states that in an AND or OR operation, the operands can be swapped without affecting the result of the operation.
The Associative law
This states that the order you group operands in for OR and AND operations doesn’t matter, so long as the original order is preserved:
The Distributive law
This states that in an AND operation, if one of the operands is an OR operation, it can be expanded by ORing the results of ANDng the value outside the brackets and the ones inside:
The Negation law
This states that double negatives cancel out:
The Redundancy law
This states that in an OR operation, the second value is redundant if it is ANDed into the first value:
De Morgan’s laws
These state that in AND / OR expressions that are all negative, you can ‘break the bar’ by splitting it and changing the operation to OR / AND: