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Topic: Closure (binary operation)

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	WULFFMAN - CTCMS
		If one were to operate the entire group on this point, a set of m equivalent points would be generated, where m is less than or equal to n, the group order.
		Closure under this operation means that for every pair of elements A and B in the group, the element C created by combining A and B under this operation (C = A * B) must also be in the group.
		The operation in the group is defined as the act of permutation.
	www.ctcms.nist.gov /wulffman/docs_1.2 (4384 words)

	Algebra Definitions
		A binary operation is an operation that combines two objects of one type to form another object of the same type.
		A set S and two operations form a field if three conditions are met: the set is a group under the first operation with commutativity, the set is a group under the second operation when the identity of the first operation is removed, and distributivity is satisfied.
		The inverse of an element a in a binary operation is the element b, which produces the identity for that operation.
	www.learner.org /channel/courses/learningmath/algebra/keyterms.html (2598 words)

	Binary operation - Wikipedia, the free encyclopedia
		Binary operations are the keystone of algebraic structures studied in abstract algebra: they form part of groups, monoids, semigroups, rings, and more.
		Typical examples of binary operations are the addition (+) and multiplication (*) of numbers and matrices as well as composition of functions on a single set.
		Binary operations are often written using infix notation such as a * b, a + b, or a · b rather than by functional notation of the form f(a,b).
	en.wikipedia.org /wiki/Binary_operation (498 words)

	Monoid
		closure: for all a, b in M, ab is in M (this is implied by the notion of binary* operation, and does not need to be required separately)
		The natural numbers with addition as the operation (identity element zero), or with multiplication as operation (identity element one).
		It is possible to view categories as generalizations of monoids: the composition of morphism in a category shares all properties of a monoid operation except that not all pairs of morphisms may be composed.
	www.ebroadcast.com.au /lookup/encyclopedia/mo/Monoid.html (626 words)

	Closure (mathematics) - Wikipedia, the free encyclopedia
		An operation of a different sort is that of finding the limit points of a subset of a topological space (if the space is first-countable, it suffices to restrict consideration to the limits of sequences but in general one must consider at least limits of nets).
		The closure is idempotent: the closure of the closure equals the closure.
		In linear algebra, the linear span of a set X of vectors is the closure of that set; it is the smallest subset of the vector space that includes X is closed under the operation of linear combination.
	en.wikipedia.org /wiki/Closure_(mathematics) (985 words)

	RELATIONAL CLOSURE:
		This concept of algebraic closure of a transformation system illustrates some of the important features of the concept we are looking for, but it is not general enough for the task of modelling complex systems by picking out all the relevant distinctions.
		This means that the complement of a closure (in the sense of complete absence of the missing elements, not in the sense of incomplete presence) can in general also be interpreted as a closure.
		Fourth, certain types of closure may be seen as generalizations or specializations of other types of closure, in the sense that a more general closure is characterized by less strict requirements, and hence is less distinction-reducing or redundancy-generating.
	pespmc1.vub.ac.be /papers/RelClosure.html (3936 words)

	Systems Operation
		An airspeed command is interpreted by the drone as a demand for a longitudinal tilt of the swash plates to produce a longitudinal thrust vector proportional to the airspeed commanded.
		This limiting system, which operates as a function of rotor rpm error, includes a frequency sensor, which provides a varying de voltage output proportional to frequency variations of the input obtained from the airborne generator.
		Depending on the operating regime (on deck or in normal flight), all or part of the follow-up signal may be applied to a washout network.
	www.gyrodynehelicopters.com /systems_operation.htm (4182 words)

	MathCog Idiocy: What is closure?
		In mathematics, the closure C(X) of an object X is defined to be the smallest object that both includes X as subset and possesses some given property.
		· The closure is idempotent: the closure of the closure equals the closure.
		To say that a set A is closed under an operation "×" means that for any members a, b of A, a×b is also a member of A. Examples: The set of all positive numbers is not closed under subtraction, since the difference of two positive numbers is in some cases not a positive number.
	mathcogidiocy.blogspot.com /2006/03/what-is-closure.html (1284 words)

	An Introduction to GROUP THEORY
		For a system to be a group the binary operation (symbolized here by "•") must be valid for any pair of elements in the group and the result of the operation must be an element of the group.
		Another example of a binary operation that is not associative is the binary operation of averaging, which I will represent as av.
		Using the closure axiom and the axiom for inverses we operate on both sides of the equation by the inverse of a.
	members.tripod.com /~dogschool/groups.html (1440 words)

	Searching for Patterns in Pascal's Triangle (Site not responding. Last check: 2007-10-10)
		An operation which takes two inputs from a given set and returns another element of the set is called a binary operation on the set.
		If a set has an element which, whenever you operate on that element and any other element of the set or on any other element of the set and that element, the operation returns the other element; then the element with this property is called the identity element for the operation.
		However, sets together with binary operations having these properties have led to an entire branch of mathematics called Group Theory, which is a subset of Abstract Algebra.
	faculty.salisbury.edu /~kmshannon/pascal/article/twist2.htm (1848 words)

	PlanetMath: groupoid
		The groupoid (or “magma”;) is closed under the operation.
		Closed binary operation by ratboy on 2004-06-04 12:13:46
		Since some groupoids of the first kind are also groupoids of the second kind, and vice versa, I think it would be beneficial to have two articles, one about groupoid (binary operation) and one about groupoid (category).
	planetmath.org /encyclopedia/Magma.html (133 words)

	College Algebra - #1 (Site not responding. Last check: 2007-10-10)
		binary operation on a set S is a rule that assigns to each ordered pair of elements of S another element in S. Example: Define x ¨ y to be equal to 2x - y for all real numbers x and y.
		This defines a binary operation on the set of real numbers since for any two numbers x and y the number x ¨ y is assigned to the pair (x, y).
		Let a, b, c, and d be elements of a field F with binary operations “+” and “´”.
	www.faculty.sfasu.edu /cproctor/1section.html (1226 words)

	Binary Operations
		It is the operation that is "binary" and it is the operation that must be associative.
		A binary operation is a function from SxS to S. The domain of the function is SxS.
		Among the axioms for a group is that it is closed under the binary operation, i.e.
	www.physicsforums.com /showthread.php?t=5708 (1148 words)

	MODELING PERMUTATION GROUP
		Continue to extend the binary operation to three elements by using the associative 2 law and then to any number of elements operating together.
		Again affirm that the patterns created by the transformations of the equilateral triangle form a set of elements which is well structured set of axioms with the the binary operations of the transformation necessary for a group D. Overhead Transparencies 1.
		Closure: Form every ordered pair a,b of elements of G the product a"O"b = c exists 3 and c is an elements of G. Associative Law: (a"O"b)"O"c = a"O"(b"O"c) 3.
	www.iit.edu /~smile/ma8604.html (529 words)

	Closure Property
		A set is closed (under an operation) if and only if the operation on two elements of the set produces another element of the set.
		Closure: When you combine any two elements of the set, the result is also included in the set.
		The elements in a binary table are displayed horizontally and vertically outside the table (in this table, the elements are 1, 2, 3, and 4).
	regentsprep.org /Regents/math/realnum/closure.htm (233 words)

	Functional programming in the Java language
		Similarly, you could pass the same closure to a different higher order function to process the data structure another way (for example, this new higher order function could implement a different logic for iterating over the constituent elements).
		In the first technique, expression specialization, a general interface for the closure is provided by the infrastructure and a concrete closure is created by writing a specialized implementation of the interface.
		As you've seen here, closures and higher order functions are not completely unfamiliar programming concepts for most Java developers, and they can be effectively combined to create a number of very handy modular solutions.
	www-128.ibm.com /developerworks/java/library/j-fp.html (4831 words)

	ABSTRACT ALGEBRA: OnLine Study Guide, Section 3.1
		Conversely, if G is a nonempty set with an associative binary operation in which the equations ax = b and xa = b have solutions for all a,b in G, then G is a group.
		Loosely, a group is a set on which it is possible to define a binary operation that is associative, has an identity element, and has inverses for each of its elements.
		This still works for the operation in a group, since if x and y are elements of a group G, and x = y, then a ·: x = a · y, for any element a in G. This is a part of the guarantee that comes with the definition of a binary operation.
	www.math.niu.edu /~beachy/abstract_algebra/study_guide/31.html (1678 words)

	Invalid Operation and Inexact Result in Control Word
		When the 80287 processor attempts a numeric operation with invalid operands or produces a result that cannot be represented, the processor will check certain numeric exceptions.
		If the result of an operation is not exactly representable in the destination format, the 80287 rounds the number and reports the precision exception.
		When projective closure is selected, the NPX treats the special values +infinity and -infinity as a single unsigned infinity.
	support.microsoft.com /kb/32865 (307 words)

	Use Closures Not Enumerations
		When you start doing this, Closure is no longer a class, but more of a MarkerInterface, because the arguments to exec depend on the structure of the Collection.
		Closures often work well when the consumer doesn't have much state (related to the enumeration) beyond what can easily be held on the stack (and to a lesser extent, in instance variables).
		Closures protect you from the details of the data structure, but that structure has to provide you with the navigation route you need (e.g.
	c2.com /cgi/wiki?UseClosuresNotEnumerations (5270 words)

	Math Forum: Ask Dr. Math FAQ: Glossary of Properties
		When you use an operation to combine an identity with another number, that number stays the same.
		An operation is commutative if you can change the order of the numbers involved without changing the result.
		They are not closed under the square root operation, because the square root of -1 is not a real number.
	mathforum.org /dr.math/faq/faq.property.glossary.html (836 words)

	Binary operation
		In the definition of a group on mathworld, http://mathworld.wolfram.com/Group.html, it is stated that the group operation is a binary operation, and it is stated that elements of a group must satisfy the four properties, including closure.
		What do you mean "isn't it sufficient to talk about the property of closure in the definition of a group?" It is not necessary to talk about the property of closure in the defintion of a group.
		Anyways, although it is not necessary, in theory, to talk about the property of closure, you are often just given a set S with a function * with domain SxS, and you have to verify that * is indeed an operation, that is, that closure does indeed hold.
	www.physicsforums.com /showthread.php?threadid=135213 (423 words)

	Maths - Group Theory - Martin Baker
		A group is any set of objects with an associated operation that combines pairs of objects in the set.
		In other words a group is defined as a set G together with a binary operation.
		Closure law: The set of objects must be closed with regard to the operation, in other words, the result of an operation must always be an element of the set.
	www.euclideanspace.com /maths/algebra/groups/index.htm (1632 words)

	Natural Numbers (Site not responding. Last check: 2007-10-10)
		Each of addition and multiplication of natural numbers is said to be a binary operation.
		Closure law for addition a + b is a unique
		Closure for multiplication a ´ b is a unique
	www.faculty.sfasu.edu /cproctor/sect32.html (250 words)

	The Citizen Scientist
		Recall that a binary operation on a set is any operation that combines any two elements of the set.
		Definition 5: - is the symbol for the binary operation of subtraction.
		Definition 9: Closure is the property that a set has when the result of the application of a particular binary operation between any two elements of set is itself an element of the set.
	www.sas.org /tcs/weeklyIssues_2006/2006-02-24/mathcorner/index.html (723 words)

	Abstraction of Groups (Site not responding. Last check: 2007-10-10)
		Technically, a group is a set of elements and an operation on the set, which takes two elements and returns a third.
		, say, isn't the set of positions of the triangle, it's the set of the motions of the triangle, and the group operation is simply that of doing one operation after another, with the result being whichever motion would have given the current position of the triangle.
		The labels s and t can applied both to positions of the triangle and to the motions which produce the positions; the set of group elements is actually the motions.
	www.cs.indiana.edu /~dasulliv/590/node3.html (245 words)

	Xah: Wallpaper: The Discontinuous Groups
		It is obvious that closure, identity, inverse, and associativity are satisfied.
		This is satisfied because in both sides of the equation, the symmetry operations are performed in the sequence A, B, C. For the case of A(BC), it merely means that we combine the operation B and C into one, and perform A followed by (B*C).
		In general, if every base point of our group elements (symmetry operations) are transformed by a rigid motion (or dilation/contraction), the resulting group is isomorphic to the original.
	xahlee.org /Wallpaper_dir/c3_Group.html (2091 words)

	A re-introduction to JavaScript - MDC
		operators use short-circuit logic, which means whether they will execute their second operand is dependent on the first.
		A closure is the combination of a function and the scope object in which it was created.
		Closures let you save state - as such, they can often be used in place of objects.
	developer.mozilla.org /en/docs/A_re-introduction_to_JavaScript (4473 words)

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