Boolean Algebra

Boolean Algebra is the branch of mathematics that works with only binary values. In contrast to numerical operations like addition, subtraction, a Boolean equation deals with a conjunction, disjunction, and negation. Boolean algebra uses binary codes to express the value and carries out logical computations through operations like AND and OR. The simplicity of Boolean expression makes it the best choice for being used in electronics and digital advancements. It would not be wrong to say that it has successfully revolutionized the world of computers. 

We shall look into Boolean expressions for logic gates and laws along with many other aspects of Boolean algebra in the following sections.

What is Boolean Algebra?

Boolean algebra works on logic instead of numbers. It recognizes only two binary values of 0 and 1. These digits hold values of:-

• True/False

• Open/Close

• Yes/No

The fundamental operators which function on Boolean algebra laws are:

• NOT: It is a unary operator, i.e., a single value. The operator is represented as A’ or ~A and exists in complements. This is the negation operation. For example ~A= 0, given A=1

• AND: It is a binary operator, hence works with two values. It can be seen as a logical version of multiplication. This operation is called the disjunction operation. Represented as A AND B or A ∧ B. 

• OR: This is another binary operator. It is the logical version of addition and is also called conjunction. Represented as: A OR B or A ∨ B.

A Boolean expression is always evaluated in terms of true or false. For instance: A AND B = 1.

A truth table can be used for all of the above operations.

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Boolean Algebra Laws 

Boolean algebra has a total of 7 laws, and they are as follows.

• Associative Law: 

The effect of logical operations is the same. Hence the sequence of performing them is irrelevant.  

• Distributive Law: 

According to the Distributive Law, multiplying two variables and combining the result with a variable produces the same value as multiplying and adding the variable with two individual variables. 

• Commutative Law: 

The commutative law in Boolean algebra states that any change in the series of variables does not affect the outcome. Expression: X.Y=Y.X

• AND Law: 

These law statements use the AND operation.

• OR Law: 

These law statements use the OR operation.

• Absorption Law: 

The Absorption Law in Boolean algebra states that A+AB= A. 

• Inversion Law: 

The inversion law states that if we use the negation operation twice, the result would be the same variable.

These Boolean algebra laws must always be kept in mind while evaluating any equation in Boolean algebra.

Boolean Expression for Logic Gates

Boolean algebra covers the concept of logic gates. These are building blocks of any circuit in the digital industry. They generally have two inputs but one output with each terminal being represented by either 0, i.e., low voltage, or 1, i.e., high voltage. Each Boolean operation has a separate gate expression. Let us see them in a little detail with Boolean algebra examples.

OR gate: The OR gate Boolean expression is A OR B. 

AND Gate: Its Boolean expression is A AND B.

NOT Gate: Its Boolean expression is A’ (A invert).

NOR Gate: The Nor gate Boolean expression is a combination of OR and Not gate. It is represented as A +B invert.

Fun Fact

Boolean algebra is named after its founder George Boole. He gave the concept in his book The Mathematical Analysis of Logic. He explained this concept in great detail in his book “An Investigation of the Laws of Thought”. Due to his contribution, he is known as the founder of computer programming.

Solved Examples: 

Q1. Convert the following Boolean equation into a truth table.

W+XY.