How to determine if a set is closed under some operation?

Question

Is the set $\{-2,0,2\}$ closed under addition? And why?

Specifically, when determining if a set is closed under an operation do you apply the operation to the each number and itself?

For example: $2+2 = 4$ would make the set not closed vs $-2+0 = -2$, $-2+2 = 0$ would make the set closed.

Answer 1

There is no notion of "set open under addition", only closed. A set is closed under some operation if applying the operation on any elements of the set gives an element which is still in that set. One counter-example is sufficient to show that the operation is not closed. $2+2 = 4 \notin \{-2,0,2\}$ shows that the operation $+$ is not closed on the set $\{-2,0,2\}$.

Hope that helps,

Answer 2

The set is not closed under addition as $2+2=4$ which is not in your set.. unless your addition is modulo $4$.