Closure Property

Jai Ganesh · 2024-06-05 22:16:04

Closure Property

Gist

Closure property states that any operation conducted on elements within a set gives a result which is within the same set of elements. Integers are either positive, negative or zero. They are whole and not fractional. Integers are closed under addition.

Summary

The closure property means that a set is closed for some mathematical operation. That is, a set is closed with respect to that operation if the operation can always be completed with elements in the set. Thus, a set either has or lacks closure with respect to a given operation.

For example, the set of even natural numbers, [2, 4, 6, 8, . . .], is closed with respect to addition because the sum of any two of them is another even natural number, which is also a member of the set. (Natural numbers are defined as the set: [1, 2, 3, 4, . . .].) It is not closed with respect to division because the quotients 6/2 and 4/8, for example, cannot be computed without using odd numbers (6/2 = 3) or fractions (4/8 = ½;), which are not members of the set.

Knowing the operations for which a given set is closed helps one understand the nature of the set. Thus, one knows that the set of natural numbers is less versatile than the set of integers because the latter is closed with respect to subtraction, but the former is not. (Integers are defined as the set: [. . .-3, -2, -1, 0, 1, 2, 3, . . .].) Similarly one knows that the set of polynomials is much like the set of integers because both sets are closed under addition, multiplication, negation, and subtraction, but are not closed under division.

Particularly interesting examples of closure are the positive and negative numbers. In mathematical structure, these two sets are indistinguishable except for one property, closure with respect to multiplication. Once one decides that the product of two positive numbers is positive, the other rules for multiplying and dividing various combinations of positive and negative numbers follow. Then, for example, the product of two negative numbers must be positive, and so on.

The lack of closure is one reason for enlarging a set. For example, without augmenting the set of rational numbers with the irrationals, one cannot solve an equation such as x2 = 2, which can arise from the use of the pythagorean theorem. Without extending the set of real numbers to include imaginary numbers, one cannot solve an equation such as x^2 + 1 = 0, contrary to the fundamental theorem of algebra.

Closure can be associated with operations on single numbers as well as operations between two numbers. When the Pythagoreans discovered that the square root of 2 was not rational, they had discovered that the rationals were not closed with respect to taking roots.

Although closure is usually thought of as a property of sets of ordinary numbers, the concept can be applied to other kinds of mathematical elements. It can be applied to sets of rigid motions in the plane, to vectors, to matrices, and to other things. For instance, one can say that the set of three-by-three matrices is closed with respect to addition.

Closure, or the lack of it, can be of practical concern, too. Inexpensive, four-function calculators rarely allow the user to use negative numbers as inputs. Nevertheless, if one subtracts a larger number from a smaller number, the calculator will complete the operation and display the negative number that results. On the other hand, if one divides 1 by 3, the calculator will display 0.333333, which is close, but not exact. If an operation takes a calculator beyond the numbers it can use, the answer it displays will be.

Details

In mathematics, a subset of a given set is closed under an operation of the larger set if performing that operation on members of the subset always produces a member of that subset. For example, the natural numbers are closed under addition, but not under subtraction: 1 - 2 is not a natural number, although both 1 and 2 are.

Similarly, a subset is said to be closed under a collection of operations if it is closed under each of the operations individually.

The closure of a subset is the result of a closure operator applied to the subset. The closure of a subset under some operations is the smallest superset that is closed under these operations. It is often called the span (for example linear span) or the generated set.

Definitions

Let S be a set equipped with one or several methods for producing elements of S from other elements of S. A subset X of S is said to be closed under these methods, if, when all input elements are in X, then all possible results are also in X. Sometimes, one may also say that X has the closure property.

The main property of closed sets, which results immediately from the definition, is that every intersection of closed sets is a closed set. It follows that for every subset Y of S, there is a smallest closed subset X of S such that

(it is the intersection of all closed subsets that contain Y). Depending on the context, X is called the closure of Y or the set generated or spanned by Y.

The concepts of closed sets and closure are often extended to any property of subsets that are stable under intersection; that is, every intersection of subsets that have the property has also the property. For example, in

a Zariski-closed set, also known as an algebraic set, is the set of the common zeros of a family of polynomials, and the Zariski closure of a set V of points is the smallest algebraic set that contains V.

In algebraic structures

An algebraic structure is a set equipped with operations that satisfy some axioms. These axioms may be identities. Some axioms may contain existential quantifiers

in this case it is worth to add some auxiliary operations in order that all axioms become identities or purely universally quantified formulas.

In this context, given an algebraic structure S, a substructure of S is a subset that is closed under all operations of S, including the auxiliary operations that are needed for avoiding existential quantifiers. A substructure is an algebraic structure of the same type as S. It follows that, in a specific example, when closeness is proved, there is no need to check the axioms for proving that a substructure is a structure of the same type.

Given a subset X of an algebraic structure S, the closure of X is the smallest substructure of S that is closed under all operations of S. In the context of algebraic structures, this closure is generally called the substructure generated or spanned by X, and one says that X is a generating set of the substructure.

For example, a group is a set with an associative operation, often called multiplication, with an identity element, such that every element has an inverse element. Here, the auxiliary operations are the nullary operation that results in the identity element and the unary operation of inversion. A subset of a group that is closed under multiplication and inversion is also closed under the nullary operation (that is, it contains the identity) if and only if it is non-empty. So, a non-empty subset of a group that is closed under multiplication and inversion is a group that is called a subgroup. The subgroup generated by a single element, that is, the closure of this element, is called a cyclic group.

In linear algebra, the closure of a non-empty subset of a vector space (under vector-space operations, that is, addition and scalar multiplication) is the linear span of this subset. It is a vector space by the preceding general result, and it can be proved easily that is the set of linear combinations of elements of the subset.

Similar examples can be given for almost every algebraic structures, with, sometimes some specific terminology. For example, in a commutative ring, the closure of a single element under ideal operations is called a principal ideal.

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#1 2024-06-05 22:16:04

Closure Property

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