Difference between revisions of "Closure"

Revision as of 23:03, 18 November 2007

Closure is a property of an abstract algebraic structure, such as a set, group, ring, or field

Definition

An algebraic structure $\mathbb{S}$ is said to have closure in a binary operation $\times$ if for any $a,b\in \mathbb{S}$ , $a\times b\in \mathbb{S}$ . In words, when any two members of $\mathbb{S}$ are combined using the operation, the result also is a member of $\mathbb{S}$ .

Examples

The real number system $\mathbb{R}$ has closure in addition, subtraction, multiplication, division, exponentiation, and also higher level operations such as $a \uparrow \uparrow b$ .
The rational number system $\mathbb{Q}$ has closure in addition, subtraction, multiplication, and division
The natural and whole number systems $\mathbb{Z}^+,\mathbb{Z}^0$ have closure in addition and multiplication.
The complex number system $\mathbb{C},\mathbb{Z}^0$ has closure in addition and subtraction.

@@ Line 7: / Line 7: @@
 *The real number system <math>\mathbb{R}</math> has closure in [[addition]], [[subtraction]], [[multiplication]], [[division]], [[exponentiation]], and also higher level operations such as <math>a \uparrow \uparrow b</math>.
 *The rational number system <math>\mathbb{Q}</math> has closure in [[addition]], [[subtraction]], [[multiplication]], and [[division]]
-*The natural and whole number system <math>\mathbb{Z}^+,\mathbb{Z}^0</math> has closure in [[addition]] and [[multiplication]].
+*The natural and whole number systems <math>\mathbb{Z}^+,\mathbb{Z}^0</math> have closure in [[addition]] and [[multiplication]].
 *The complex number system <math>\mathbb{C},\mathbb{Z}^0</math> has closure in [[addition]] and [[subtraction]].

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Difference between revisions of "Closure"

Revision as of 23:03, 18 November 2007

Definition

Examples

See Also