Does closure property work in reverse? That is, if $a + b$ belongs to a set closed under addition, are $a$ and $b$ members of that set?

Question

If a set is closed under addition, and a number $a + b$ belongs to that set, does it follow that $a$ and $b$ are members of that set?

Answer 1

No: the even numbers are closed under addition and $1 + 1$ is even, but $1$ is not even.