Difference between revisions of "Closure"
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==Examples== | ==Examples== | ||
| − | *The real number system <math>\mathbb{R}</math> has closure in [[addition]], [[subtraction]], [[multiplication]], [[division]], [[ | + | *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]]. | *The natural and whole number system <math>\mathbb{Z}^+,\mathbb{Z}^0</math> has closure in [[addition]] and [[multiplication]]. | ||
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 
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 system
has closure in addition and multiplication. - The complex number system
has closure in addition and subtraction.
has closure in 
.
has closure in 
has closure in 
has closure in 
