Closure

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 and subtraction.
• The integral number system has closure in addition, subtraction, multiplication, and exponentiation.

See Also

• Set theory
• Ring theory
• Group theory
• Field theory