- Boolean Algebra Definition: Boolean algebra is a branch of mathematics focused on variables valued either 1 or 0, used primarily in digital circuit design.
- Core Operations: It revolves around three fundamental operations—AND, OR, and NOT—to handle logical operations in binary systems.
- Theorems and Laws: Boolean algebra includes critical theorems like De Morgan’s, which simplify the conversion between ANDs to ORs and vice versa, using complementation.
- Logical Diagram Representation: Expressions in Boolean algebra can be depicted through various logic gates, aiding in understanding circuit designs.
- Practical Application: Boolean algebra is essential for creating and simplifying digital circuits, proving its utility with each theorem and law.
What is Boolean Algebra?
Boolean algebra is a different kind of algebra or rather can be said a new kind of algebra which was invented by world-famous mathematician George Boole in the year of 1854. He published it in his book “An Investigation of the Laws of Thought”.
Later using this technique Claude Shannon introduced a new type of algebra which is termed as Switching Algebra. In digital electronics, there are several methods of simplifying the design of logic circuits. This algebra is one of these methods.
George Boole introduced symbols to represent the structure of logical thoughts. This type of algebra follows specific rules, known as the laws of Boolean algebra, which govern logical operations.
There are several key theorems in Boolean algebra that simplify and speed up calculations. Boolean logic uses just two variables, 1 and 0, to perform all operations.
Boolean algebra or switching algebra is a system of mathematical logic used for operations in a binary system, utilizing three basic operations: AND, OR, and NOT.
There are many rules in Boolean algebra by which those mathematical operations are done. In Boolean algebra, the variables are represented by English Capital Letter like A, B, C, etc and the value of each variable can be either 1 or 0, nothing else.
In Boolean algebra, expressions can be transformed into logic diagrams using various logic gates such as AND gate, OR gate, NOT gate, NOR gates, NAND gates, XOR gates, and XNOR gates.
Some basic logical Boolean operations,
AND Operation
OR Operation
Not Operation
Some basic laws for Boolean Algebra
A . 0 = 0 where A can be either 0 or 1.
A . 1 = A where A can be either 0 or 1.
A . A = A where A can be either 0 or 1.
A . Ā = 0 where A can be either 0 or 1.
A + 0 = A where A can be either 0 or 1.
A + 1 = 1 where A can be either 0 or 1.
A + Ā = 1
A + A = A
A + B = B + A where A and B can be either 0 or 1.
A . B = B . A where A and B can be either 0 or 1.
The laws of Boolean algebra are also true for more than two variables like,
Cumulative Law for Boolean Algebra
According to Cumulative Law, the order of OR operations and AND operations conducted on the variables makes no differences.
Associative Laws for Boolean Algebra
This law is for several variables, where the OR operation of the variables result is the same through the grouping of the variables. This law is quite the same in the case of AND operators.
Distributive Laws for Boolean Algebra
This law is composed of two operators, AND and OR.
Let us show one use of this law to prove the expression 
Proof:
Redundant Literal Rule
From the truth table,
| Inputs | Output | ||
| A | B | ĀB | A + ĀB |
| 0 | 0 | 0 | 0 |
| 0 | 1 | 1 | 1 |
| 1 | 0 | 0 | 1 |
| 1 | 1 | 0 | 1 |
| Inputs | Output | |
| A | B | A+B |
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 1 |
The truth table demonstrates that,
Absorption Laws for Boolean Algebra
Proof from truth table,
| Inputs | Output | ||
| A | B | AB | A+A.B |
| 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 1 |
| 1 | 1 | 1 | 1 |
Both A and A+A.B column is the same.
Proof from truth table,
| A | B | A+B | A.X(A+B) |
| 0 | 0 | 0 | 0 |
| 0 | 1 | 1 | 0 |
| 1 | 0 | 1 | 1 |
| 1 | 1 | 1 | 1 |
Both A and A.X or A(A+B) columns are the same.
From De Morgan’s Theorem,
Proof from truth table,
Examples of Boolean Algebra
These are another method of simplifying a complex Boolean expression. In this method, we only use three simple steps.
- Complement entire Boolean expression.
- Change all ORs to ANDs and all ANDs to ORs.
- Now, complement each of the variables and get the final expression.
By this method,
will be first complemented, i.e.
.Now, change all (+) to (.) and (.) to (+) i.e.
Now, complement each of the variables,
This is the final simplified form of a Boolean expression,
And it is exactly equal to the results which have been come by applying De Morgan Theorem.
Another example,
By the Second Method,
Representation of Boolean function in the truth table.
Let us consider a Boolean function,
Now let us represent the function in the truth table.
Thus we have shown some basic laws of Boolean algebra. On the other page, we have described De Morgan’s theorems and related laws on it.
