Boolean Algebra is a branch of mathematics that plays a significant role in digital logic and computer science. Developed by George Boole in the 19th century, it is particularly used to express logical propositions and operations in a mathematical form. Unlike classical arithmetic, Boolean Algebra operates not with numbers and standard operations, but with truth values (typically represented as “true” and “false” or 1 and 0) and logical operations.

2.1. Fundamental Structure of Boolean Algebra

1. Variables and Values:

In Boolean algebra, variables can only take one of two values: 1 (true) or 0 (false).

This principle forms the foundation of digital electronics in the operation of computers.

2. Basic Logical Operations:
a. AND Operation (∧): The result is 1 if and only if all values involved are 1; otherwise, the result is 0. For example, the expression ( A \land B ) is true if both A and B are true.
b. OR Operation (∨): The result is 1 if at least one of the values is 1; otherwise, the result is 0. For example, the expression ( A \lor B ) is true if either A or B is true.
c. NOT Operation (¬): This operation inverts the value. For instance, ( \neg A ) represents the opposite of A; if A is true, the result is false, and if A is false, the result is true.

These fundamental operations are used to create more complex logical expressions and circuits. Below, these expressions are presented in tabular form.

A	B	A ∧ B
0	0	0
0	1	0
1	0	0
1	1	1

A	B	AvB
0	0	0
0	1	1
1	0	1
1	1	1

A	¬A
0	1
1	0

After discussing Boolean Algebra above, we can now make the following definition: Using multiple Boolean variables (A, B, C, …), we can combine these variables with logical operations to create Boolean expressions. These logical operations are called logical operators.

The logical operators referred to here are the AND, OR, and NOT operations, for which a table was provided earlier.

These logical operators follow certain laws, similar to set algebra. For instance:

• The ∨ operation is analogous to the union (∪) operation in set algebra.
• The ∧ operation is analogous to the intersection (∩) operation.

These laws indicate that Boolean expressions and set operations can be manipulated in similar ways. We will discuss these in the next section.