22 Facts About Boolean logic

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

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

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

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

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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 logic circuit relates time complexity to circuit complexity.

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

The Duality Principle, called De Morgan duality, asserts that Boolean logic algebra is unchanged when all dual pairs are interchanged.

For so-called "active-high" Boolean logic, 0 is represented by a voltage close to zero or "ground", while 1 is represented by a voltage close to the supply voltage; active-low reverses this.

Certainly any law satisfied by all concrete Boolean logic algebras is satisfied by the prototypical one since it is concrete.

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.

Boolean logic algebras are special here, for example a relation algebra is a Boolean logic algebra with additional structure but it is not the case that every relation algebra is representable in the sense appropriate to relation algebras.

We say that Boolean logic algebra is finitely axiomatizable or finitely based.

Many syntactic concepts of Boolean algebra carry over to propositional logic with only minor changes in notation and terminology, while the semantics of propositional logic are defined via Boolean algebras in a way that the tautologies of propositional logic correspond to equational theorems of Boolean algebra.

Entailment differs from implication in that whereas the latter is a binary operation that returns a value in a Boolean logic 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.

Today, all modern general purpose computers perform their functions using two-value Boolean logic; that is, their electrical circuits are a physical manifestation of two-value Boolean logic.

However much of a straitjacket this might prove in practice for the respondent, the principle of the simple yes-no question has become a central feature of both judicial and mathematical Boolean logic, making two-valued Boolean logic deserving of organization and study in its own right.

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

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