# Boolean Logic System

A Boolean Logic System is a Deductive Logic System that involves Boolean Logic Operations.

**AKA:**Boolean Logic, Boolean Logic Theory, Boolean Logic Calculus, Boolean Algebra.**Context:**- It can have Boolean Logic Symbol.

**See:**Formal Language, Propositional Logic System.

## References

### 2013

- (Wikipedia, 2013) ⇒ http://en.wikipedia.org/wiki/Boolean_algebra_(logic)
- In mathematics and mathematical logic,
**Boolean algebra**is the subarea of algebra in which the values of the variables are the truth values*true*and*false*, usually denoted 1 and 0 respectively. Instead of elementary algebra where the values of the variables are numbers, and the main operations are addition and multiplication, the main operations of Boolean algebra are the conjunction*and*, denoted ∧, the disjunction*or*, denoted ∨, and the negation*not*, denoted ¬.

- In mathematics and mathematical logic,

- http://en.wikipedia.org/wiki/Boolean_algebra_%28structure%29#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

*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*= 0complements

- 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 Boolean algebra with only one element is called a

*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**.

- The relation ≤ defined by

- http://en.wikipedia.org/wiki/Boolean_algebra#Basic_operations
- The basic operations of Boolean algebra are the following ones:
- And (conjunction), denoted
*x*∧*y*(sometimes*x*AND*y*or K*xy*), satisfies*x*∧*y*= 1 if*x*=*y*= 1 and*x*∧*y*= 0 otherwise. - Or (disjunction), denoted
*x*∨*y*(sometimes*x*OR*y*or A*xy*), satisfies*x*∨*y*= 0 if*x*=*y*= 0 and*x*∨*y*= 1 otherwise. - Not (negation), denoted ¬
*x*(sometimes NOT*x*, N*x*or !*x*), satisfies ¬*x*= 0 if*x*= 1 and ¬*x*= 1 if*x*= 0.

- And (conjunction), denoted
- If the truth values 0 and 1 are interpreted as integers, these operation may be expressed with the ordinary operations of the arithmetic:
*x*∧*y*=*xy*,*x*∨*y*=*x*+*y*-*xy*,- ¬
*x*= 1 -*x*.

- Alternatively the values of
*x*∧*y*,*x*∨*y*, and ¬*x*can be expressed by tabulating their values with truth tables as follows.

- The basic operations of Boolean algebra are the following ones:

*x**y**x*∧*y**x*∨*y*0 0 0 0 1 0 0 1 0 1 0 1 1 1 1 1 *x*¬ *x*0 1 1 0 **Figure 1. Truth tables**### 2009

- (WordNet, 2009) ⇒ http://wordnetweb.princeton.edu/perl/webwn?s=boolean%20logic
- S: (n) Boolean logic, Boolean algebra (a system of symbolic logic devised by George Boole; used in computers)

- http://en.wiktionary.org/wiki/Boolean_algebra
- Noun
- 1. (algebra, logic, computing) An algebra in which all elements can take only one of two values (typically 0 and 1, or "true" and "false") and are subject to operations based on AND, OR and NOT

- Noun
- http://en.wiktionary.org/wiki/Boolean_logic
- Noun
- 1. (mathematics, logic, computing) A system of symbolic logic that is the basis of Boolean algebra

- Noun
- http://www.noblenet.org/merrimack/guides/infolit_glossary.htm
- Use of operators AND, OR, NOT to combine search terms. (see KEYWORDS/KEYWORD SEARCHING).

- http://mason.gmu.edu/~montecin/netterms.htm
- a type of logic (using AND, OR, NOT operators, for example) used by search engines to find information on the Internet and in electronic databases. (For example, to find computer viruses instead of human viruses, you might try the keywords "computers and viruses.")

- (Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/Boolean_logic
- Boolean logic is a complete system for logical operations. It was named after George Boole, who first defined an algebraic system of logic in the mid 19th century. Boolean logic has many applications in electronics, computer hardware and software, and is the base of digital electronics. In 1938, Claude Shannon showed how electric circuits with relays were a model for Boolean logic. This fact soon proved enormously consequential with the emergence of the electronic computer.
- Using the algebra of sets, this article contains a basic introduction to sets, Boolean operations, Venn diagrams, truth tables, and Boolean applications. The Boolean algebra article discusses a type of algebraic structure that satisfies the axioms of Boolean logic. The binary arithmetic article discusses the use of binary numbers in computer systems.
- Terms: Let X be a set:
- An
**element**is one member of a set. This is denoted by \in. If it's not an element of the set, this is denoted by \notin. - The
**universe**is the set X, sometimes denoted by 1. Note that this use of the word universe means "all elements being considered", which are not necessarily the same as "all elements there are". - The
**empty set**or null set is the set of no elements, denoted by \varnothing and sometimes 0. - A
**unary operator**applies to a single set. There is one unary operator, called logical NOT. It works by taking the complement. - A
**binary operator**applies to two sets. The basic binary operators are logical OR and logical AND. They perform the union and intersection of sets. There are also other derived binary operators, such as XOR (exclusive OR). - A
**subset**is denoted by A \subseteq B and means every element in set A is also in set B. - A
**superset**is denoted by A \supseteq B and means every element in set B is also in set A. - The
**identity**or equivalence of two sets is denoted by A \equiv B and means that every element in set A is also in set B and every element in set B is also in set A. - A
**proper subset**is denoted by A \subset B and means every element in set A is also in set B and the two sets are not identical. - A
**proper superset**is denoted by A \supset B and means every element in set B is also in set A and the two sets are not identical.

- An

- (WordNet, 2009) ⇒ http://wordnetweb.princeton.edu/perl/webwn?s=boolean%20logic