I want to prove the following statements:
- Is the function "sin" definable in the structure $(\Bbb{R},<,+,\cdot,0,1)$, that is does there exists a formula $\phi=\phi(x_0,x_1)$ such that for all $a,b\in\Bbb{R}$: $sin(a)=b$ iff $(\Bbb{R},<,+,\cdot,0,1)$ realizes $\phi[a,b]$ ?!
- Is $\Bbb{Z}$ a definable subset of $(\Bbb{R},<,+,\cdot,0,1)$ ?
- Is $\Bbb{Q}$ a definable subset of $(\Bbb{R},<,+,\cdot,0,1)$ ?
I have no idea how to solve this questions. I think i have to prove that i have to show that in a extension of the structure the subsets are not definable or i have to give a formula. Can someone help me?! Thanks