英和和英

名詞

free Boolean algebra (複数形 free Boolean algebras)

1. (algebra) A field of sets whose elements are equivalent to Boolean formulas (または, 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.