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Boolean algebra
名詞
(algebra)
An
algebraic
structure
(𝛴,∨,∧,∼,0,1)
where
∨
and
∧
are
idempotent
binary
operators,
∼
is
a
unary
involutory
operator
(called
"complement"),
and
0
and
1
are
nullary
operators
(i.e.,
constants),
such
that
(𝛴,∨,0)
is
a
commutative
monoid,
(𝛴,∧,1)
is
a
commutative
monoid,
∧
and
∨
distribute
with
respect
to
each
other,
and
such
that
combining
two
complementary
elements
through
one
binary
operator
yields
the
identity
of
the
other
binary
operator.
(See
Boolean
algebra
(structure)#Axiomatics.)
日本語の意味
ブール代数：論理演算（論理和、論理積、補集合など）により定義される代数的構造。集合や論理回路、計算理論などで用いられ、0や1を含む定数や、分配法則、冪等性など特有の公理系に基づいた体系である。
関連語
free Boolean algebra
名詞
(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
2ⁿmembers
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
2^(2ⁿ)
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.
日本語の意味
自由ブール代数とは、n個の独立な生成子（変数）から自由に構成されるブール代数であり、各要素がブール公式（またはその同値類）に対応する集合の体系として定義される。 / 具体的には、まずn個の生成子から原子（各原子は各変数の真偽値の割り当て、またはその対応する条件）を構成し、その原子の集合の冪集合をとることで、2^(2ⁿ)個の要素を持つ集合体としてブール代数が表現される。
関連語
