Common Boolean Algebra Rules

Boolean algebra rules vary from typical algebra rules in ways that can seem confounding. Boolean math can't be solved in the same way as standard algebra, so you'll need to understand what the basic rules are telling you about the nature of the problem and its goal.

Know Your Operators
In Boolean math, operators have very different meanings that require specific actions to find a solution. Operators are sometimes represented in different ways, depending on where they appear in the equation. If you have questions about how to interpret the operators, consult your source material to determine which representation your source is using for each operator.

• AND: In Boolean algebra, the AND operator says that the output is on if all of the inputs are on. If any of the inputs are not on, the output is not on.
• OR: The OR operator says that if any input is on, the output is on. This means that if some inputs are on and some inputs are off, the output is still on.
• NAND: The NAND operator says that if all inputs are on, the output is off. If any of the inputs are off, the output is on.
• NOR: The NOR operator says that if any input is on, the output is off. The output is only on if all inputs are off.
• XOR: Also known as Exclusive OR, this states that if the inputs are different, the output is on.
• NOT: This inverter operation says that the input and output are in opposite states. For example, if an input is on, the output is off.

Truth Tables
Truth tables are one of the messiest ways to solve Boolean math. When you lay out a truth table, you spell out every possible outcome for a given operation and input. Then, you look at the operation, inputs and outputs to determine the outcome. If possible, use Boolean algebra to solve these problems; otherwise, truth tables can take a lot of time and space to work out.

Boolean Algebra Theorems
Knowing several common Boolean algebra theorems can make your life infinitely easier. Learn these theorems to easily recognize and solve Boolean algebra problems: OR identity, AND identity, double negation, commutativity of OR, commutativity or AND, associativity of OR, associativity of AND, distributivity and DeMorgan's Theorem. By learning these common theorems, you'll be able to solve Boolean algebra problems without resorting to truth tables.

Unary Operators Take Precedence
If you're dealing with both binary and unary operators, the unary expressions take precedence. Deal with the unary operators first, and then turn your attention to the binary operators.