Boolean Algebra

  • Boolean algebra is a mathematical structure used for logical operations, named after George Boole.

  • Boolean algebra consists of a set, two binary operations (logical AND and logical OR), unary operation (NOT), and two elements (0 and 1).

  • Binary operator ‘AND’, often denoted as a dot, works such that if both inputs are TRUE (or 1), the result is TRUE; otherwise, it is FALSE (or 0).

  • Binary operator ‘OR’, often represented as a plus, works such that if at least one input is TRUE, the result is TRUE; if both inputs are FALSE, the result is FALSE.

  • Unary operator ‘NOT’, signified by an overbar, a prime or an exclamation mark, inverts the truth value. If the input is TRUE, the result is FALSE and vice versa.

  • In Boolean algebra, laws of commutativity, associativity, and distributivity all apply. Commutativity law states that the order of inputs does not affect result for AND and OR operations. Associativity law implies that the grouping does not affect the operation result. Distributive law means that AND distributes over OR and vice versa.

  • Other important principles include the identity law (an element ANDed with 0 or ORed with 1 results in the element itself), null law (an element ANDed with 0 gives 0 and ORed with 1 gives 1), complement law (an element ANDed with its complement gives 0; ORed with its complement gives 1), and the absorption law (an element ORed with (element AND another element) results in the original element and vice versa).

  • Understanding DeMorgan’s theorems is crucial—they state that the complement of a conjunction is equivalent to the disjunction of the complements; and the complement of a disjunction is equivalent to the conjunction of the complements.

  • Sum-of-Products (SOP) and Product-of-Sums (POS) are two standard forms of expressing Boolean functions. SOP is a logical expression in which individual terms are logically ANDed, and these resultant terms are ORed. POS is an expression in which individual terms are ORed, and these resultant terms are ANDed.

  • Karnaugh maps are graphical representations of Boolean functions useful in simplifying operations. They minimize the possibility of human errors and make it easier to spot patterns for simplification.

  • Boolean algebra is critical in computer science as it corresponds directly to the operations of binary digital logic gates in circuits, helping in circuit design and optimization. It also forms the basis of designing and simplifying logic circuits like adders, subtractors, multiplexers, etc.