Two-element Boolean algebra

https://en.wikipedia.org/wiki/Two-element_Boolean_algebra

The two-element Boolean algebra is the Boolean algebra whose underlying set (or universe or carrier) B is the Boolean domain, B = {0, 1}. Paul Halmos's name for this algebra "2" is also common.

Definition

B is a partially ordered set and the elements of B are also its bounds.

An operation of arity n is a mapping from Bⁿ to B, Bⁿ ⟼ B

Boolean algebra consists of two binary operations and unary complementation.

The binary operations have been named and notated in various ways. Here they are called 'sum' and 'product', and notated by infix '+' and '∙', respectively.

Sum and product commute and associate, as in the usual algebra of real numbers.

As for the order of operations, brackets are decisive if present. Otherwise '∙' precedes '+'. Hence A∙B + C is parsed as (A∙B) + C and not as A∙(B + C).

Complementation is denoted by writing an overbar over its argument (or a tick). The numerical analog of the complement of X is 1 − X.

In the language of universal algebra, a Boolean algebra is algebra denoted by the 6-tuple ⟨B, +, ∙, ◌', 1, 0⟩, and has the type ⟨2,2,1,0,0⟩.

$\langle\ B\ ,\ +\ ,\ \cdot\ ,\ \overline {..}\ , \ 1\ ,\ 0\ \rangle$

Either one-to-one correspondence between {0,1} and {True,False} yields classical bivalent logic in equational form, with complementation read as NOT. If 1 is read as True, + is read as OR, and ∙ as AND (and vice versa if 1 is read as False). These two operations define a commutative semiring, known as the Boolean semiring.