Definition
A Boolean algebra is a six-tuple consisting of a set A, equipped with two binary operations ∧ (called "meet" or "and"), ∨ (called "join" or "or"), a unary operation ¬ (called "complement" or "not") and two elements 0 and 1 (sometimes denoted by the symbols ⊥ and ⊤, respectively), such that for all elements a, b and c of A, the following axioms hold:
-
-
a ∨ (b ∨ c) = (a ∨ b) ∨ c a ∧ (b ∧ c) = (a ∧ b) ∧ c associativity a ∨ b = b ∨ a a ∧ b = b ∧ a commutativity a ∨ 0 = a a ∧ 1 = a identity a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c) a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c) distributivity a ∨ ¬a = 1 a ∧ ¬a = 0 complements
-
A Boolean algebra with only one element is called a trivial Boolean algebra or a degenerate Boolean algebra. (Some authors require 0 and 1 to be distinct elements in order to exclude this case.)
It follows from the last three pairs of axioms above (identity, distributivity and complements) that
-
- a = b ∧ a if and only if a ∨ b = b.
The relation ≤ defined by a ≤ b if and only if the above equivalent conditions hold, is a partial order with least element 0 and greatest element 1. The meet a ∧ b and the join a ∨ b of two elements coincide with their infimum and supremum, respectively, with respect to ≤.
As in every bounded lattice, the relations ∧ and ∨ satisfy the first three pairs of axioms above; the fourth pair is just distributivity. Since the complements in a distributive lattice are unique, to define the involution ¬ it suffices to define ¬a as the complement of a.
The set of axioms is self-dual in the sense that if one exchanges ∨ with ∧ and 0 with 1 in an axiom, the result is again an axiom. Therefore by applying this operation to a Boolean algebra (or Boolean lattice), one obtains another Boolean algebra with the same elements; it is called its dual.
Read more about this topic: Boolean Algebra (structure)
Famous quotes containing the word definition:
“The physicians say, they are not materialists; but they are:MSpirit is matter reduced to an extreme thinness: O so thin!—But the definition of spiritual should be, that which is its own evidence. What notions do they attach to love! what to religion! One would not willingly pronounce these words in their hearing, and give them the occasion to profane them.”
—Ralph Waldo Emerson (1803–1882)
“Beauty, like all other qualities presented to human experience, is relative; and the definition of it becomes unmeaning and useless in proportion to its abstractness. To define beauty not in the most abstract, but in the most concrete terms possible, not to find a universal formula for it, but the formula which expresses most adequately this or that special manifestation of it, is the aim of the true student of aesthetics.”
—Walter Pater (1839–1894)
“The definition of good prose is proper words in their proper places; of good verse, the most proper words in their proper places. The propriety is in either case relative. The words in prose ought to express the intended meaning, and no more; if they attract attention to themselves, it is, in general, a fault.”
—Samuel Taylor Coleridge (1772–1834)