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	The Mathematics of Boolean Algebra (Stanford Encyclopedia of Philosophy)
		Boolean algebra is the algebra of two-valued logic with only sentential connectives, or equivalently of algebras of sets under union and complementation.
		Another general algebraic notion which applies to Boolean algebras is the notion of a free algebra.
		Much of the deeper theory of Boolean algebras, telling about their structure and classification, can be formulated in terms of certain functions defined for all Boolean algebras, with infinite cardinals as values.
	plato.stanford.edu /entries/boolalg-math (2063 words)

	[No title] (Site not responding. Last check: 2007-10-15)
		The two-element Boolean algebra is also important in the general theory of Boolean algebras, because an equation involving several variables is generally true in all Boolean algebras if and only if it is true in the two-element Boolean algebra (which can always be checked by a trivial brute force algorithm).
		Every Boolean algebra (A, ∧, ∨) gives rise to a ring (A, +, ) by defining a + b = (a ∧ ¬b) ∨ (b ∧ ¬a) (this operation is called "symmetric difference" in the case of sets and XOR in the case of logic) and a b = a ∧ b.
		An ideal of the Boolean algebra A is a subset I such that for all x, y in I we have x ∨ y in I and for all a in A we have a ∧ x in I.
	wikiwhat.com /encyclopedia/b/bo/boolean_algebra.html (1570 words)

	Boolean algebra
		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 algebra and order theory.
		Every Boolean algebra (A,,) gives rise to a ring (A, +, ) by defining a + b = (a ¬b) (b ¬a) (this operation is called "symmetric difference" in the case of sets and XOR in the case of logic) and a b = a b.
		An ideal of the Boolean algebra A is a subset I such that for all x, y in I we have x y in I and for all a in A we have a x in I.
	www.knowledgefun.com /book/b/bo/boolean_algebra.html (1657 words)

	PlanetMath: Boolean lattice
		Given a set, any collection of subsets that is closed under unions, intersections, and complements is a Boolean algebra.
		Boolean rings (with identity, but allowing 0=1) are equivalent to Boolean lattices.
		This is version 4 of Boolean lattice, born on 2002-02-24, modified 2004-02-18.
	planetmath.org /encyclopedia/BooleanAlgebra2.html (132 words)

	Boolean algebra - HighBeam Encyclopedia (Site not responding. Last check: 2007-10-15)
		BOOLEAN ALGEBRA [Boolean algebra], an abstract mathematical system primarily used in computer science and in expressing the relationships between sets (groups of objects or concepts).
		When used in set theory, Boolean notation can demonstrate the relationship between groups, indicating what is in each set alone, what is jointly contained in both, and what is contained in neither.
		Boolean algebra is of significance in the study of information theory, the theory of probability, and the geometry of sets.
	www.encyclopedia.com /doc/1E1-booleanal.html (325 words)

	Boolean algebra
		Stone was initially interested in Boolean algebras in order to gain insight into the structure of rings of functions which were being investigated in functional analysis.
		In the 1930's Stone proved that every Boolean algebra is isomorphic to a field of sets, and that the equations true of the two-element Boolean algebra are the same as the equations true of all Boolean algebras; and these equations were consequences of a small initial set of defining equations.
		Boolean algebra became a deep and fascinating subject in its own right, with much more to offer than a convenient notation to analyze simple chains of reasoning.
	www.math.uwaterloo.ca /~snburris/htdocs/scav/boolean/boolean.html (573 words)

	Boolean Algebra - Search Results - MSN Encarta
		Boolean Algebra, branch of mathematics having laws and properties similar to, but different from, those of ordinary algebra.
		Digital logic is a rational process for making simple “true” or “false” decisions based on the rules of Boolean algebra.
		- algebra concerned with binary combinations: a form of algebra concerned with the logical functions of variables that are restricted to two values, true or false.
	uk.encarta.msn.com /Boolean_Algebra.html (116 words)

	Boolean Algebra
		I was struck by the number of folks with little understanding of Boolean algebra, the basis for the design of logic circuits.
		I was fascinated to learn that Boolean algebra is an important trick used in the defense of philosophical ideas.
		Boolean algebra, and the tools we use to deal with it, can help simplify, or at least document, such convoluted code.
	www.ganssle.com /articles/aboolea.htm (1967 words)

	Boolean Algebra
		I was struck by the number of folks with little understanding of Boolean algebra, the basis for the design of logic circuits.
		I was fascinated to learn that Boolean algebra is an important trick used in the defense of philosophical ideas.
		Boolean algebra, and the tools we use to deal with it, can help simplify, or at least document, such convoluted code.
	www.avocetsystems.com /company/articles/magazine/aboolea.htm (1900 words)

	Boolean algebra
		Boolean logic, or Boolean algebra as it is called today, was developed by an English mathematician, George Boole, in the 19th century.
		Boolean algebra also enables the engineers to achieve the desired output by using the fewest number of logic gates.
		Q.34 Boolean algebra is used primarily by _______ to simplifty circuits.
	www.tpub.com /neets/book13/54h.htm (617 words)

	Defining Boolean Algebra (Site not responding. Last check: 2007-10-15)
		And the * in Boolean Algebra doesn't "multiply" but "mutually filters out" to form a set that gives you what is common in the sets that were "combined" in such a manner.
		The first postulate in regular algebra means that whatever you do with two numbers, you will end up with another number that belongs to the set of all possible numbers.
		Well, in Boolean Algebra, when applied to binary numbers, there is only two numbers, 0 and 1, and if you are limited to only one digit each, then whatever operation you do on the two numbers, you will get one of the two numbers.
	www.jairosoft.com /intro_math13.htm (1979 words)

	Boolean Algebra
		Boolean algebra provides the ability to work with many interesting functions while often using significantly less space than a truth table.
		We will informally prove that boolean algebra can describe all boolean functions - this justifies our use of gates since they correspond to the building blocks of boolean algebra.
		Boolean expressions and circuits are interchangeable though circuits are generally smaller since they may share common subexpressions.
	cs-people.bu.edu /jconsidi/teaching/notes/cs210/node6.html (342 words)

	Boolean Searching
		The Boolean OR command is used in order to allow any of the specified search terms to be present on the web pages listed in results.
		Boolean searching can only be done from the advanced search page, as listed on the Search Assistance Features page.
		Lycos says it supports many Boolean commands, and I haven't verified these, because of the difficulty of determining exactly which datasets might be processed.
	searchenginewatch.com /showPage.html?page=2155991 (1159 words)

	Introduction - Chapter 7: BOOLEAN ALGEBRA - Volume IV - Digital (Site not responding. Last check: 2007-10-15)
		Just bear in mind that the system of numbers defining Boolean algebra is severely limited in terms of scope, and that there can only be one of two possible values for any Boolean variable: 1 or 0.
		Once you comprehend the premise of all quantities in Boolean algebra being limited to the two possibilities of 1 and 0, and the general philosophical principle of Laws depending on quantitative definitions, the "nonsense" of Boolean algebra disappears.
		The difference is that Boolean quantities are restricted to a single bit (either 1 or 0), whereas binary numbers may be composed of many bits adding up in place-weighted form to a value of any finite size.
	www.allaboutcircuits.com /vol_4/chpt_7/1.html (947 words)

	Internet Web Search, Boolean Algebra Logic
		The concept of boolean algebra is embedded in human psychology, in our very biological understanding of how the world works.
		Boolean algebra is very similar, with the logical operators AND, OR, and NOT, combined with operands that can have either a value of True or False in expressions like the following:
		Boolean algebra queries are supported by most search engines, but sometimes only through the "advanced" version.
	www.livinginternet.com /w/wu_expert_bool.htm (533 words)

	Boolean Algebra
		Boolean Algebra is both a formalization of the algebraic aspects of logic, and the customary language of logic used by the designers of computers.
		An important aspect of the axioms and properties of Boolean Algebra (and therefore of logic as well) is the notion of "duality".
		In general, given any axiom or property in Boolean Algebra, if we interchange the sum and product operators and at the same time interchange the elements 0 and 1, the new expression will also be a valid property of the algebra.
	www.rwc.uc.edu /koehler/comath/24.html (892 words)

	M567: Boolean Algebra
		of ideals of the Boolean algebra B is not a Boolean algebra.
		Show that a subset of a Boolean algebra is an ideal of the Boolean algebra if and only if it is an ideal of the corresponding Boolean ring.
		Show that a subset of a Boolean algebra is a prime ideal of the Boolean algebra if and only if it is a prime ideal of the corresponding Boolean ring.
	orion.math.iastate.edu /jdhsmith/class/M567S05.htm (880 words)

	COROLLARY THEOREMS - ELECTRONINC DESIGN NOTES: Boolean Algebra
		As logic mathematics branch, algebra was brought to us by the Arab migratory people in the seventh century; more specifically, it is derived from Al-Jabar (Al-Jabar means reunion), the title of a book written by the great mathematician Muhammad Ibn Musa Al-Khwarizmi in 820.
		Boolean Algebra is a Mathematical Model developed for the TRUE and FALSE states, by Mr.
		In Boolean Algebra "=" means "may be substituted with" not "equal to".
	www.corollarytheorems.com /Design/boolean.htm (1031 words)

	Boolean Algebra
		Leibniz initiated the search for a system of symbols with rules of their combination in his De Arte Combinatoria of 1666, as well as developing the binary notation.
		Boolean algebra and the design of all modern binary digital computers has depended on the results of this work.
		He subsequently produced works on The Fifth Book of Euclid proved Algebraically and A Syllabus of Plan Algebraical Geometry as well as collections of algebraic and arithmetic formulae to aid examination candidates.
	vmoc.museophile.org /algebra/section3_4.html (263 words)

	Amazon.com: Schaum's Outline of Boolean Algebra and Switching Circuits: Books: Elliott Mendelson (Site not responding. Last check: 2007-10-15)
		The treatment here of Boolean algebra, deeper than in most elementary texts, can serve as a supplement or an introduction to graduate-level study.
		This book is devoted to two separate and related topics: the theory of Boolean algebra and logic and also the synthesis and simplification of switching and logic circuits.
		Chapter 3 discusses Boolean algebra, which is a set B together with two binary operations, a singular operation, the two specific elements 0 and 1, and a set of axioms.
	amazon.com /Outline-Boolean-Algebra-Switching-Circuits/dp/0070414602 (1499 words)

	Boolean Algebra (Site not responding. Last check: 2007-10-15)
		The simpler expression can then be implemented with a smaller, simpler circuit, which in turn saves the price of the unnecessary gates, reduces the number of gates needed, and reduces the power and the amount of space required by those gates.
		One tool to reduce logical expressions is the mathematics of logical expressions, introduced by George Boole in 1854 and known today as Boolean Algebra.
		The rules of Boolean Algebra are simple and straight-forward, and can be applied to any logical expression.
	www.play-hookey.com /digital/boolean_algebra.html (340 words)

	Boolean algebra
		A transformation from a Boolean equation control specification to a Petri net.
		State preferences and international institutions: Boolean analysis of China's use of force and South China sea territorial disputes....
		Parametric Estimation of a Boolean Segment Process With Stochastic Restoration Estimation.
	www.infoplease.com /ce6/sci/A0808301.html (327 words)

	Boolean Algebra
		The expression x < y is a boolean expression; it returns true if the value stored in x is less than the value stored in y.
		The operands for relational operands are typically numeric (int or float), but we also use them to compare characters.
		Logical operators take boolean operands and return a boolean result.
	www.eecs.wsu.edu /~cs150/boolean_algebra.htm (170 words)

	Boolean algebra
		there are a number of useful unary and binary operations on booleans, including
		we can generalize the structure of the boolean truth values and associated operations by axiomatizing some of their salient properties (associative, commutative, distributive laws, etc.): the result is an abstract algebra called a boolean algebra
		in particular, the subsets of some base set form a boolean algebra, where various operations on sets play the role of the boolean operations
	www.willamette.edu /~fruehr/446/lectures/review5.html (122 words)

	Digital Logic - Gates, Boolean Algebra
		The algebra is unusual because the variables in the algebra (S, P, C and W in the example) can take only two values, 0 and 1.
		Boolean Algebra can be a confusing and misleading business.
		If you have a Boolean function that is a sum-of-products form it can be implemented using a two layer circuit with the first layer composed of AND gates, and the second layer composed of OR gates.
	www.facstaff.bucknell.edu /mastascu/eLessonsHTML/Logic/Logic1.html (3330 words)

	Boolean Algebra (Site not responding. Last check: 2007-10-15)
		Leibniz initiated the search for a system of symbols with rules of their combination in his De Arte Combinatoria of 1666, as well as developing the binary notation.
		Boolean algebra and the design of all modern binary digital computers has depended on the results of this work.
		He subsequently produced works on The Fifth Book of Euclid proved Algebraically and A Syllabus of Plan Algebraical Geometry as well as collections of algebraic and arithmetic formulae to aid examination candidates.
	vmoc.museophile.com /algebra/section3_4.html (263 words)

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