Boolean Algebra, Boolean Expression, Rules and Theorems

Boolean algebra is a branch of algebra that only uses ‘false’ and ‘true’ values for variables and is usually denoted by 0 and 1. It is defined as a system of logic where variables are represented as whole numbers between 0 and 1. The values that you take on the numbers are true or false, but not both at the same time. A variable is either true or false, but never both true and false simultaneously.

The variables of Boolean algebra can take only one of two possible values, zero and one. Still, like any other mathematical expression, Boolean expressions too can have an infinite number of variables, all of which represent different individual inputs to the expression. Take an example where the variables P, Q, and R form a logical expression such as P + Q = R. Although each of these variables can only have one value, either a zero or a one. This means that all operations within Boolean algebra are based on only two things: true and false.

History

There are certain laws and rules in Boolean algebra to define the operations of digital logic circuits. To fully understand these laws, you must first understand Boolean algebra. It is a type of binary logic developed by George Boole in 1854 and published two years later. To use this new maths correctly, you must study its history and why it came into existence. Certain rules, laws and theorems of Boolean algebra were invented to help reduce the number of logic gates needed in performing a particular logic operation.

Boolean Algebra Laws and Examples

A complete understanding of the laws and theorems must be grasped to use Boolean algebra properly.

Boolean expressions can also be converted using logic gates like OR gate, AND gate, NOT gate, NOR gates, XOR gates, XNOR gates, NAND gates, etc.

The three basic Boolean operations are: 

1. OR gate returns ‘true’ or ‘1’ if either of the input variables is true.

2. AND gate returns ‘true’ or ‘1’ only if all the input variables are true.

3. NOT gate returns the complement value of the input variable.

Basic Laws of Boolean Algebra

In all the cases given below A can either be 0 or 1

1. A . 0 = 0 
2. A . 1 = A 
3. A . A = A 
4. A . Ā = 0 
5. A + 0 = A 
6. A + 1 = 1 
7. A + Ā = 1
8. A + A = A

Some basic Boolean algebra laws that are used to simplify Boolean expressions are:

1. Idempotent Law

A * A = A

A + A = A

2. Associative Law

(A * B) * C = A * (B * C)

(A + B) + C = A + (B + C)

3. Commutative Law

A * B = B * A

A + B = B + A

4. Distributive Law

A * (B + C) = A * B + A * C

A + (B * C) = (A + B) * (A + C)

5. Identity Law

A * 0 = 0     A * 1 = A

A + 1 = 1     A + 0 = A

6. Complement Law

A * ~A = 0

A + ~A = 1

7. Involution Law

~(~A) = A

8. DeMorgan’s Law

~(A * B) = ~A + ~B

~(A + B) = ~A * ~B

9. Absorption

A + (A * B) = A

A * (A + B) = A

(A * B) + (A * ~B) = A

(A + B) * (A + ~B) = A

A + (~A * B) = A + B

A * (~A + B) = A * B

Conclusion

Boolean algebra laws are a series of laws and theorems that help understand why or how something happens with Boolean algebra, which is essential to any computer program.

The main goal of logic design is to simplify the logic as much as possible so that the final implementation is easy. To simplify the logic, you need to simplify the Boolean equations and expressions that represent the logic. Some laws and theorems have been proposed to simplify the Boolean equations. Using these laws and theorems makes it very easy to simplify or reduce the logical complexity of a Boolean expression or function.