# is this set closed under addition?

I have some revision questions in my maths books and I'm a bit stuck on this one.

Is $S=\{n^2:n \in \mathbb{Z}\}$ closed under the usual addition.

I know that for it to be closed the sum of any 2 elements of $S$ must also be in $S$

The solution says it isn't closed but doesn't say why, could anyone help me with this?

Much appreciated

• You just find a counterexample. $2^2+1^2=5$ and $5$ is not a square. – Hanul Jeon Oct 25 '14 at 10:30
• @tetori Thanks, that is super obvious when you explain it like that! – panderson Oct 25 '14 at 10:38

Does $1^2+2^2 \in S$? ${}{}{}{}{}$