Closed under

A set has closure under an operation if performance of that operation on members of the set always produces a member of the same set; in this case we also say that the set is closed under the operation. For example, the positive integers are closed under addition, but not under subtraction: ${\displaystyle 1-2}$ is not a positive integer even though both 1 and 2 are positive integers. Another example is the set containing only zero, which is closed under addition, subtraction and multiplication (because ${\displaystyle 0+0=0}$, ${\displaystyle 0-0=0}$, and ${\displaystyle 0\times {0}=0}$).

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