# Closure

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 $\in\$ A then a * b $\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
Some counter-examples: