Generated with

Bool

algebra.lattice.Bool

See theBool companion object

trait Bool[A] extends Heyting[A] with GenBool[A]

Boolean algebras are Heyting algebras with the additional constraint that the law of the excluded middle is true (equivalently, double-negation is true).

This means that in addition to the laws Heyting algebras obey, boolean algebras also obey the following:

(a ∨ ¬a) = 1
¬¬a = a

Boolean algebras generalize classical logic: one is equivalent to "true" and zero is equivalent to "false". Boolean algebras provide additional logical operators such as xor, nand, nor, and nxor which are commonly used.

Every boolean algebras has a dual algebra, which involves reversing true/false as well as and/or.

Attributes

Companion:: object
Source:: Bool.scala
Graph
Supertypes: trait GenBool[A]
trait Heyting[A]
trait BoundedDistributiveLattice[A]
trait DistributiveLattice[A]
trait BoundedLattice[A]
trait BoundedJoinSemilattice[A]
trait BoundedMeetSemilattice[A]
trait Lattice[A]
trait MeetSemilattice[A]
trait JoinSemilattice[A]
trait Serializable
class Any
Known subtypes: class BooleanAlgebra
class BoolFromBoolRing[A]
class DualBool[A]
Self type: Bool[A]

Members list

Concise view

Value members

Concrete methods

Every Boolean algebra is a BoolRing, with multiplication defined as and and addition defined as xor. Bool does not extend BoolRing because, e.g. we might want a Bool[Int] and CommutativeRing[Int] to refer to different structures, by default.

Note that the ring returned by this method is not an extension of the Rig returned from BoundedDistributiveLattice.asCommutativeRig.

Attributes

Definition Classes: GenBool
Source:: Bool.scala

This is the lattice with meet and join swapped

Attributes

Definition Classes: BoundedDistributiveLattice -> BoundedLattice -> Lattice
Source:: Bool.scala

Attributes

Source:: Bool.scala

The operation of ''relative complement'', symbolically often denoted a\b (the symbol for set-theoretic difference, which is the meaning of relative complement in the lattice of sets).