Boolean algebras abstract the algebra of sets.
This later became known as a Boolean data type .
., true and false), uses the logic of Boolean algebra.
"A good boolean search might include 'project manag* AND PM,' but making sure to check for those discrepancies will help a job seeker be found without relying on the recruiter's boolean skills.
Hence every Boolean algebra is a distributive lattice.
Since a Boolean algebra is a poset, it is also a category.
Even the theory of Boolean algebras with a distinguished ideal is decidable.
To define this we need to introduce the idea of a Boolean-valued model of set theory.
The methods of modern algebra began to be applied to Boolean algebra in the 20th century.
Justification Logic starts with the simplest base: classical Boolean logic, and for good reasons.
The concept of goal functions is particularly prominent in the logical framework of Boolean games (Harrenstein 2004).
An almost-definitive property of the class of Boolean algebras is that their polynomials in the initial Boolean algebra are all the operations on that algebra.
Much of the deeper theory of Boolean algebras, telling about their structure and classification, can be formulated in terms of certain functions defined for all Boolean algebras, with infinite cardinals as values.
So far we have considered two types of complexity facts may have: Boolean complexity (which in turn divides into the various types corresponding to the various Boolean operations) and the complexity of substantial facts.
The strong no-conspiracy condition requires that any Boolean combination of choices of measurement directions in the two wings of the EPR experiment are probabilistically independent of any Boolean combination of the hypothetical common causes.
Not long after this he discovered a translation between Boolean algebras and Boolean rings; under this translation the ideals of a Boolean algebra corresponded precisely to the ideals of the associated Boolean ring.
Another central topic is duality: Boolean algebras are dual to Stone spaces, complete atomic Boolean algebras are dual to sets, distributive lattices with top and bottom are dual to partially ordered sets, algebraic lattices are dual to semilattices, and so on.
The category Set is a topos; so also are (i) the category Set(B) of Boolean-valued sets and mappings in any Boolean extension V(B) of the universe of sets; (ii) the category of sheaves of sets on a topological space; (iii) the category of all diagrams of maps of sets
Since the set of events corresponding to all projector operators on a given Hilbert space does not have a Boolean structure, the Born probability (which is defined over these projectors) does not satisfy the definition of probability of Kolmogorov (which applies to a Boolean algebra of events).
We say that two similar structures A, B are Boolean isomorphic, written A ≅b B, if, for some complete Boolean algebra B, we have V(B) ⊨ A ≅ B, that is, if there is a Boolean extension of the universe of sets in which the canonical representatives of A and B are isomorphic with Boolean value 1.
- of or relating to a combinatorial system devised by George Boole that combines propositions with the logical operators AND and OR and IF THEN and EXCEPT and NOT
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