hi

would you help me find out if those sets are closed under addition and scalar product or not.

$\displaystyle A_{1}=\left \{ (cosx,sinx)/x\in \mathbb{R} \right \}$

$\displaystyle A_{2}=\left \{ (x,y)/x\in \mathbb{Z},y\in \mathbb{Z} \right \}$

for the second i think $\displaystyle \mathbb{Z}$ is closed under addition,$\displaystyle \left (\forall x\in \mathbb{Z} \right ),\left ( \forall y\in \mathbb{Z} \right ); \left (x+y \right )\in \mathbb{Z}$.am i right ?

thanks for helping me.