Boolean Algebra

Abstract

Boolean algebra is a set $\mathbb {B}=\{0,1\}$ with defined on it operations of disjunction ($\vee$), conjunction ($\wedge$) and negation ($\overline{ {a}}$). Boolean variable p can take the values 0 or 1 (and only them), $p\in \mathbb {B}$.

Boolean expression is formed by Boolean variables with the help of operations $\vee$, $\wedge$, $\overline{ {a}}$ and brackets. The expression of the form $x_1\wedge x_2$ is sometimes written in the form $x_1$ &   $x_2$ or simply $x_1 x_2$.

The tuple $\mathbf {x}=(x_1,x_2,\dots , x_n)$, where $x_i\in \{0,1\}$, $1\leqslant i\leqslant n$, is referred to the Boolean tuple (vector). The tuple elements are components or coordinates. The function $f:\mathbb {B}^n\rightarrow \mathbb {B}$, where $f(x_1,\dots , x_n)$ is Boolean expression, is called Boolean function.

The most important classes of Boolean functions are considered:

• Functions that preserve constants 0 and 1.

• Self-dual functions.

• Linear functions.

• Monotone functions.

For practical applications, an important task consists in reducing the Boolean function written in disjunctive normal form. One of the methods to solve this problem was developed by Karnaugh.

The methods of Boolean algebra are used for creation and analysis of combinational circuits, that allow understanding the structure and the logic of digital devices’ operation, including computers. Combinational circuits consist of electronic devices (called gates) with a finite number of inputs and outputs, and each input and output can have only two values of the signal.