Relative complement
In
mathematics, a
lattice L is said to be
relatively complemented if for all
a,
b,
c in
L with
a ≤
b ≤
c there is some
x in
L such that
x ∨
b =
c and
x ∧
b =
a. An element
x with this property is a
relative complement of
b in the interval [
a,
c].
Two particular cases are frequently seen:
then the complement of A relative to B (the interval involved is from the empty set to B) is
- If the lattice is a Boolean algebra, then the complement of b relative to the integral [a, c] is a ∨ (~ b) ∧ c. (In general, the expression x ∨ y ∧ z is ambiguous in Boolean algebra. But the fact that a is sufficient for b and c is necessary for b removes the ambiguity in this case.) In the usual interpretation of Boolean algebra as a model of propositional logic, if a is a sufficient condition for b and c is a necessary condition for b, the complement of b relative to the interval [a, c] is the unique (up to logical equivalence) proposition d such that
- a is sufficient for d an c is necessary for d, and
- d becomes equivalent to [not b] if one learns that a is false and c is true.
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