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They are named after George Boole, an English mathematician at University College Cork, who first defined them as part of a system of logic in the mid 19th century. Specifically, Boolean algebra was an attempt to use algebraic techniques to deal with expressions in the propositional calculus. Today, Boolean algebras find many applications in electronic design. They were first applied to switching by Claude Shannon in the 20th century19th century 20th century 21st century more centuries) Decades: 1900s 1910s 1920s 1930s 1940s 1950s 1960s 1970s 1980s 1990s As a means of recording the passage of time, the 20th century was that century which lasted from 1901- 2000 in the sense of the Gre.

The operators of Boolean algebra may be represented in various ways. Often they are simply written as AND, OR and NOT. In describing circuits, NAND (NOT AND), NOR (NOT OR) and XOR (exclusive OR) may also be used. Mathematicians often use + for OR and . for AND (since in some ways those operations are analogous to addition and multiplication in other algebraic structures) and represent NOT by a line drawn above the expression being negated.

Here we use another common notation with (or ^ for browsers that don't support the character) for AND, (or v) for OR, and ¬ (or ~) for NOT.

1 Definition and first consequences

A Boolean algebra is a latticeSee lattice for other mathematical as well as non-mathematical meanings of the term. In mathematics, a lattice is a partially ordered set in which all nonempty finite subsets have both a supremum join and an infimum meet . On the other hand, lattices can (A, , ) (considered as an algebraic structure) with the following four additional properties:

1. bounded below: There exists an element 0, such that a 0 = a for all a in A.
2. bounded above: There exists an element 1, such that a 1 = a for all a in A.
3. distributive law: For all a, b, c in A, (a b) c = (a c) (b c).
4. existence of complements: For every a in A there exists an element ¬a in A such that a ¬a = 1 and a ¬a = 0.

From these axioms, one can directly show that the smallest element 0, the largest element 1, and the complement ¬a of any element a are uniquely determined.

Like any lattice, a Boolean algebra (A, , ) gives rise to a partially ordered set (A, ≤) by defining

a ≤ b iffIn mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if . It is often, not always, written italicized: iff''. Although "P iff Q" is most standard, common alternative phrases include "P a = a b

(which is also equivalent to b = a b).

In fact one can also define a Boolean algebra to be a distributive lattice (A, ≤) (considered as a partially ordered set) with least element 0 and greatest element 1, within which every element x has a complement ¬x such that

x ¬x = 0 and x ¬x = 1

Here and are used to denote the infimumIn mathematics the infimum of a subset of some set is the greatest element that is smaller than all other elements of the subset. Consequently the term greatest lower bound is also commonly used. Infima of real numbers are a common special case that is es (meet) and supremumIn mathematics, the supremum of a given set is the least element which is greater than or equal to each element of the set. Consequently, it is also referred to as the least upper bound . In general, unless a set contains a greatest element, the supremum (join) of two elements. Again, if complements in the above sense exist, then they are uniquely determined.

The algebraic and the order theoretic perspective as usually can be used interchangeably and both are of great use to import results and concepts from both universal algebraUniversal algebra is the field of mathematics that studies the ideas common to all algebraic structures. Basic idea From the point of view of universal algebra, an algebra is a set A together with a collection of operations on A. An n- ary operation on A and order theoryOrder theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of a mathematical ordering. This article gives a detailed introduction to the field and includes some of the most basic definitions.. In many practical examples an ordering relation, conjunction, disjunction, and negation are all naturally available, so that it is straightforward to exploit this relationships.

Now one can obtain several other theorems valid in all Boolean algebras. For example, for all elements a and b of a Boolean algebra, one finds that

• a 0 = 0 and a 1 = 1,
• ¬1 = 0 and ¬0 = 1,
• ¬ ¬ a = a

and that both de Morgan's laws are valid, i.e.

• ¬(a b) = (¬a) (¬b) and ¬(a b) = (¬a) (¬b)

One can also apply general insights from duality in order theory to Boolean algebras. Especially, the order dual of every Boolean algebra, or, equivalently, the algebra obtained by exchanging and , is also a Boolean algebra. Thus the dual version of the distributive law,

• (a b) c = (a c) (b c)

also holds true. In general, any law valid for Boolean algebras can be transformed into another valid dual law by exchanging 0 with 1, with , and ≤ with ≥.