20 Facts About Boolean algebra

In mathematics and mathematical logic, Boolean algebra is the branch of algebra in which the values of the variables are the truth values true and false, usually denoted 1 and 0, respectively.

Boolean algebra was introduced by George Boole in his first book The Mathematical Analysis of Logic, and set forth more fully in his An Investigation of the Laws of Thought.

Boolean algebra has been fundamental in the development of digital electronics, and is provided for in all modern programming languages.

Precursor of Boolean algebra was Gottfried Wilhelm Leibniz's algebra of concepts.

Boole's Boolean algebra predated the modern developments in abstract Boolean algebra and mathematical logic; it is however seen as connected to the origins of both fields.

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Efficient implementation of Boolean algebra functions is a fundamental problem in the design of combinational logic circuits.

Boolean algebra is not sufficient to capture logic formulas using quantifiers, like those from first order logic.

The closely related model of computation known as a Boolean algebra circuit relates time complexity to circuit complexity.

Three Boolean algebra operations described above are referred to as basic, meaning that they can be taken as a basis for other Boolean algebra operations that can be built up from them by composition, the manner in which operations are combined or compounded.

Law of Boolean algebra is an identity such as = ? z between two Boolean terms, where a Boolean term is defined as an expression built up from variables and the constants 0 and 1 using the operations ?, ?, and ¬.

The empty set and X This two-element algebra shows that a concrete Boolean algebra can be finite even when it consists of subsets of an infinite set.

From this bit vector viewpoint, a concrete Boolean algebra can be defined equivalently as a nonempty set of bit vectors all of the same length and closed under the bit vector operations of bitwise ?, ?, and ¬, as in 1010?0110 = 0010,1010?0110 = 1110, and ¬1010 = 0101, the bit vector realizations of intersection, union, and complement respectively.

Conversely any law that fails for some concrete Boolean algebra must have failed at a particular bit position, in which case that position by itself furnishes a one-bit counterexample to that law.

However, if we represent each divisor of n by the set of its prime factors, we find that this nonconcrete Boolean algebra is isomorphic to the concrete Boolean algebra consisting of all sets of prime factors of n, with union corresponding to least common multiple, intersection to greatest common divisor, and complement to division into n So this example while not technically concrete is at least "morally" concrete via this representation, called an isomorphism.

Entailment differs from implication in that whereas the latter is a binary operation that returns a value in a Boolean algebra, the former is a binary relation which either holds or does not hold.

The natural interpretation of is as = in the partial order of the Boolean algebra defined by x = y just when x?y = y This ability to mix external implication and internal implication ? in the one logic is among the essential differences between sequent calculus and propositional calculus.

Claude Shannon formally proved such behavior was logically equivalent to Boolean algebra in his 1937 master's thesis, A Symbolic Analysis of Relay and Switching Circuits.

Original application for Boolean algebra operations was mathematical logic, where it combines the truth values, true or false, of individual formulas.

In view of the highly idiosyncratic usage of conjunctions in natural languages, Boolean algebra cannot be considered a reliable framework for interpreting them.

Boolean algebra operations are used in digital logic to combine the bits carried on individual wires, thereby interpreting them over {0,1}.