free Boolean algebra

English

Noun

free Boolean algebra (plural free Boolean algebras)

(algebra) A field of sets whose elements are equivalent to Boolean formulas (or, perhaps more precisely, equivalence classes of Boolean formulas). Starting with a set of n variables which are independent of each other and are called generators, the power set of this set has ${\\displaystyle 2^{n}}$ members which may be called atoms and are valuations of the n variables: a valuation can be considered to be a set of variables which are "true" under that valuation, or a conjunction of generators (such that variables not included in that set are included in negated form in the equivalent conjunction). Then the power set of the set of atoms yields a set of ${\\displaystyle 2^{2^{n}}}$ members which are the elements of the said field of sets. These elements correspond to Boolean formulas: a formula can be considered to be a set of valuations which make the formula true, or a linear combination (i.e., a disjunction) of atoms.

References

2017 October 11, Brian Huffman, “Free Boolean Algebra”, in Archive of Formal Proofs‎, retrieved 2018-3-4:

单词	free Boolean algebra
释义	free Boolean algebra English Noun free Boolean algebra (plural free Boolean algebras) (algebra) A field of sets whose elements are equivalent to Boolean formulas (or, perhaps more precisely, equivalence classes of Boolean formulas). Starting with a set of n variables which are independent of each other and are called generators, the power set of this set has ${\\displaystyle 2^{n}}$ members which may be called atoms and are valuations of the n variables: a valuation can be considered to be a set of variables which are "true" under that valuation, or a conjunction of generators (such that variables not included in that set are included in negated form in the equivalent conjunction). Then the power set of the set of atoms yields a set of ${\\displaystyle 2^{2^{n}}}$ members which are the elements of the said field of sets. These elements correspond to Boolean formulas: a formula can be considered to be a set of valuations which make the formula true, or a linear combination (i.e., a disjunction) of atoms. References 2017 October 11, Brian Huffman, “Free Boolean Algebra”, in Archive of Formal Proofs‎, retrieved 2018-3-4:
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