Difference between revisions of "Closure"

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*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 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 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]].
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*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]].
 
*The complex number system <math>\mathbb{C},\mathbb{Z}^0</math> has closure in [[addition]] and [[subtraction]].
  

Revision as of 00:03, 19 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

See Also