## Most recent change of Closure

Edit made on November 24, 2008 by ColinWright at 17:05:21

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A Set is closed under a binary operation if any two elements of the set when combined using the binary operation produce an element of the same set.

Thus a set A is closed under the binary operation * if for all a and b EQN:\in\ A then a * b EQN:\in\ A.

This idea extends beyond simple binary operations.

For example:

* The set of natural numbers is closed under addition and multiplication

** but not closed under subtraction.

* The set of the integers is closed under addition, subtraction and multiplication

** but not under division

The set of natural numbers is closed under addition but not closed under subtraction.

Some counter-examples:

* The set of rational numbers is not closed under minimum upper bound

* The set of real numbers is not closed under square roots