**BOOLEAN ALGEBRA**

**WHAT IS THE BOOLEAN ALGEBRA?**

It is a special branch of algebra that is used mainly in digital electronics. Boolean algebra was invented in the year 1854 by the English mathematician George Boole.

Boolean algebra is a method to simplify logic circuits (or sometimes called logic switching circuits) in digital electronics.

Therefore, it is also called as “Change of algebra”. We can represent the operation of logic circuits using numbers, following some rules, which are well known as “Laws of Boolean algebra”.

We can also do the calculations and logical operations of the circuits even faster by following some theorems, which are known as “Boolean algebra theorems”. A Boolean function is a function that represents the relationship between the input and the output of a logic circuit.

Boolean logic only allows two circuits states, such as true and false. These two states are represented by 1 and 0, where 1 represents the “true” state and 0 represents the “false” state.

The most important thing to remember in Boolean algebra is that it is very different from regular mathematical algebra and its methods. Before learning about Boolean algebra, let’s talk a little about the history of Boolean algebra and its invention and development.

**HISTORY OF THE BOOLE ALGEBRA**

As mentioned earlier, Boolean algebra was invented in the year 1854 by the English mathematician George Boole. First he declared the idea of Boole’s algebra in his book “An investigation of the laws of thought”.

After this, the Boolean algebra is well known as the perfect way to represent the digital logic circuits.

At the end of the 19th century, scientist Jevons, Schoroder y Huntington used this concept for modernized terms. In the year 1936, MHStone showed that the Boolean algebra is ‘isomofphic’ for the sets (a functional area in mathematics).

In the 1930s, a scientist named Claude Shannon developed a new method of algebra type “Algebra change” using the concepts of Boolean algebra, to study the switching circuits.

The logical synthesis of modern electronic automation tools is represented efficiently by using Boolean functions known as “binary decision diagrams”.

Boolean algebra allows only two states in a logic circuit, such as true and false, high and low, yes or no, open and close or 0 and 1.

**LAWS AND IDENTITIES OF THE BOOLEAN ALGEBRA**

When formulating mathematical expressions for logic circuits it is important to have knowledge of Boolean algebra, which defines the rules for expressing and simplifying binary logical statements. A bar on a symbol indicates the Boolean operation NOT, which corresponds to the inversion of a signal.

**Fundamental laws**

**Commutative laws**

*A + B = B + A*

A ∙ B = B ∙ A

A ∙ B = B ∙ A

**Associative laws**

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

(A ∙ B) ∙ C = A ∙ (B ∙ C)

(A ∙ B) ∙ C = A ∙ (B ∙ C)

**Distributive laws**

*A ∙ (B + C) = (A ∙ B) + (A ∙ C)*

A + (B ∙ C) = (A + B) ∙ (A + C)

A + (B ∙ C) = (A + B) ∙ (A + C)

**Other useful identities**

*A + (A ∙ B) = A*

A ∙ (A + B) = A

A + (A ∙ B) = A + B

(A + B) ∙ (A + B) = A

(A + B) ∙ (A + C) = A + (B ∙ C)

A + B + (A ∙ B) = A + B

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

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

A ∙ (A + B) = A

A + (A ∙ B) = A + B

(A + B) ∙ (A + B) = A

(A + B) ∙ (A + C) = A + (B ∙ C)

A + B + (A ∙ B) = A + B

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

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

**SIMPLIFICATION OF BOOLEAN FUNCTIONS**

By using Boolean theorems and laws, we can simplify Boolean expressions, by which we can reduce the required number of logic gates to be implemented. We can simplify the Boolean function using two methods:

- The algebraic method: through the use of identities (Boolean laws).
- The graphic method: using the Karnaugh Map method.