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 systems <math>\mathbb{ | + | *The natural and whole number systems <math>\mathbb{N}^+,\mathbb{N}^0</math> have closure in [[addition]] and [[multiplication]]. |
| − | *The complex number system <math>\mathbb{C}</math> has closure in [[addition]] and [[subtraction]]. | + | *The complex number system <math>\mathbb{C}</math> has closure in [[addition]], [[subtraction]], [[multiplication]], [[division]], [[exponentiation]], and also higher level operations such as <math>a \uparrow \uparrow b</math>. |
| + | *The [[integer|integral]] number system <math>\mathbb{Z}</math> has closure in [[addition]], [[subtraction]], [[multiplication]], and [[exponentiation]]. | ||
==See Also== | ==See Also== | ||
Latest revision as of 20:55, 20 August 2008
Closure is a property of an abstract algebraic structure, such as a set, group, ring, or field
Definition
An algebraic structure 
is said to have closure in a binary operation 
if for any 
, 
. In words, when any two members of are combined using the operation, the result also is a member of .
Examples
- The real number system
has closure in addition, subtraction, multiplication, division, exponentiation, and also higher level operations such as 
. - The rational number system
has closure in addition, subtraction, multiplication, and division - The natural and whole number systems
have closure in addition and multiplication. - The complex number system
has closure in addition, subtraction, multiplication, division, exponentiation, and also higher level operations such as . - The integral number system
has closure in addition, subtraction, multiplication, and exponentiation.
has closure in 
.
has closure in 
have closure in 
has closure in 
has closure in 
