In mathematics and mathematical logic, Boolean algebra is the branch 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 as ∧, the disjunction or denoted as ∨, and the negation not denoted as ¬. It is thus a formalism for describing logical relations in the same way that ordinary algebra describes numeric relations.
Boolean algebra was introduced by George Boole in his first book The Mathematical Analysis of Logic (1847), and set forth more fully in his An Investigation of the Laws of Thought (1854).[1] According to Huntington, the term "Boolean algebra" was first suggested by Sheffer in 1913.[2]
Boolean algebra has been fundamental in the development of digital electronics, and is provided for in all modern programming languages. It is also used in set theory and statistics.[3]
History[]
Boole's algebra predated the modern developments in abstract algebra and mathematical logic; it is however seen as connected to the origins of both fields.[4] In an abstract setting, Boolean algebra was perfected in the late 19th century by Jevons, Schröder, Huntington, and others until it reached the modern conception of an (abstract) mathematical structure.[4] For example, the empirical observation that one can manipulate expressions in the algebra of sets by translating them into expressions in Boole's algebra is explained in modern terms by saying that the algebra of sets is a Boolean algebra (note the indefinite article). In fact, M. H. Stone proved in 1936 that every Boolean algebra is isomorphic to a field of sets.
In the 1930s, while studying switching circuits, NEC engineer Akira Nakashima independently discovered Boolean algebra, which he was unaware of until 1938. In a series of papers published from 1934 to 1936, he formulated a two-valued Boolean algebra as a way to analyze and design circuits by algebraic means in terms of logic gates.[5][6][7][8] Nakashima's work was later cited and elaborated on in Claude Shannon's seminal 1938 paper "A Symbolic Analysis of Relay and Switching Circuits".[7] Shannon observed that one could apply the rules of Boole's algebra in this setting, for which he coined the term switching algebra. Shannon already had at his disposal the abstract mathematical apparatus, thus he cast switching algebra as the two-element Boolean algebra. In circuit engineering settings today, there is little need to consider other Boolean algebras, thus "switching algebra" and "Boolean algebra" are often used interchangeably.[9][10][11] Efficient implementation of Boolean functions is a fundamental problem in the design of combinational logic circuits. Modern electronic design automation tools for VLSI circuits often rely on an efficient representation of Boolean functions known as (reduced ordered) binary decision diagrams (BDD) for logic synthesis and formal verification.[12]
See also[]
- Binary number
- Boolean algebra (structure)
- Boolean algebras canonically defined
- Booleo
- Heyting algebra
- Intuitionistic logic
- List of Boolean algebra topics
- Logic design
- Propositional calculus
- Relation algebra
- Switching circuit theory
- Three-valued logic
- Vector logic
References[]
- ↑ Boole, George (2003) [1854]. An Investigation of the Laws of Thought. Prometheus Books. ISBN 978-1-59102-089-9.
- ↑ "The name Boolean algebra (or Boolean 'algebras') for the calculus originated by Boole, extended by Schröder, and perfected by Whitehead seems to have been first suggested by Sheffer, in 1913." E. V. Huntington, "New sets of independent postulates for the algebra of logic, with special reference to Whitehead and Russell's Principia mathematica", in Trans. Amer. Math. Soc. 35 (1933), 274-304; footnote, page 278.
- ↑ Givant, Steven; Halmos, Paul (2009). Introduction to Boolean Algebras. Undergraduate Texts in Mathematics, Springer. ISBN 978-0-387-40293-2.
- ↑ 4.0 4.1 J. Michael Dunn; Gary M. Hardegree (2001). Algebraic methods in philosophical logic. Oxford University Press US. p. 2. ISBN 978-0-19-853192-0.
- ↑ History of Research on Switching Theory in Japan, IEEJ Transactions on Fundamentals and Materials, Vol. 124 (2004) No. 8, pp. 720-726, Institute of Electrical Engineers of Japan
- ↑ Switching Theory/Relay Circuit Network Theory/Theory of Logical Mathematics, IPSJ Computer Museum, Information Processing Society of Japan
- ↑ 7.0 7.1 Radomir S. Stanković (University of Niš), Jaakko T. Astola (Tampere University of Technology), Mark G. Karpovsky (Boston University), Some Historical Remarks on Switching Theory, 2007, DOI 10.1.1.66.1248
- ↑ Radomir S. Stanković, Jaakko Astola (2008), Reprints from the Early Days of Information Sciences: TICSP Series On the Contributions of Akira Nakashima to Switching Theory, TICSP Series #40, Tampere International Center for Signal Processing, Tampere University of Technology
- ↑ Norman Balabanian; Bradley Carlson (2001). Digital logic design principles. John Wiley. pp. 39–40. ISBN 978-0-471-29351-4., online sample
- ↑ Rajaraman & Radhakrishnan. Introduction To Digital Computer Design An 5Th Ed. PHI Learning Pvt. Ltd. p. 65. ISBN 978-81-203-3409-0.
- ↑ John A. Camara (2010). Electrical and Electronics Reference Manual for the Electrical and Computer PE Exam. www.ppi2pass.com. p. 41. ISBN 978-1-59126-166-7.
- ↑ Shin-ichi Minato, Saburo Muroga (2007). "Binary Decision Diagrams". In Wai-Kai Chen (ed.). The VLSI handbook (2nd ed.). CRC Press. ISBN 978-0-8493-4199-1. chapter 29.
Further reading[]
- Mano, Morris; Ciletti, Michael D. (2013). Digital Design. Pearson. ISBN 978-0-13-277420-8.
- J. Eldon Whitesitt (1995). Boolean algebra and its applications. Courier Dover Publications. ISBN 978-0-486-68483-3. Suitable introduction for students in applied fields.
- Dwinger, Philip (1971). Introduction to Boolean algebras. Würzburg: Physica Verlag.
- Sikorski, Roman (1969). Boolean Algebras (3/e ed.). Berlin: Springer-Verlag. ISBN 978-0-387-04469-9.
- Bocheński, Józef Maria (1959). A Précis of Mathematical Logic. Translated from the French and German editions by Otto Bird. Dordrecht, South Holland: D. Reidel.
Historical perspective
- George Boole (1848). "The Calculus of Logic," Cambridge and Dublin Mathematical Journal III: 183–98.
- Theodore Hailperin (1986). Boole's logic and probability: a critical exposition from the standpoint of contemporary algebra, logic, and probability theory (2nd ed.). Elsevier. ISBN 978-0-444-87952-3.
- Dov M. Gabbay, John Woods, ed. (2004). The rise of modern logic: from Leibniz to Frege. Handbook of the History of Logic. Vol. 3. Elsevier. ISBN 978-0-444-51611-4., several relevant chapters by Hailperin, Valencia, and Grattan-Guinesss
- Calixto Badesa (2004). The birth of model theory: Löwenheim's theorem in the frame of the theory of relatives. Princeton University Press. ISBN 978-0-691-05853-5., chapter 1, "Algebra of Classes and Propositional Calculus"
- Burris, Stanley, 2009. The Algebra of Logic Tradition. Stanford Encyclopedia of Philosophy.
- Radomir S. Stankovic; Jaakko Astola (2011). From Boolean Logic to Switching Circuits and Automata: Towards Modern Information Technology. Springer. ISBN 978-3-642-11681-0.
External links[]
- Boolean Algebra chapter on All About Circuits
- How Stuff Works – Boolean Logic
- Science and Technology - Boolean Algebra contains a list and proof of Boolean theorems and laws.