I have the following task and I am not so sure about my solution:
a) Is $\{0\}$ definable over $(\mathbb{R}, +, \cdot)$?
b) Is $\{1\}$ definable over $(\mathbb{R}, +, \cdot)$?
c) Is $<$ definable over $(\mathbb{R}, +, \cdot)$?
So I have to give a formula, using only the symbols of the modell, for $\{0\}$, $\{1\}$ and $<$.
a) My try is the following:
$\varphi\equiv\forall v_0(v_0+v_1=v_0)$
This should be only true, if $v_1=0$
b)
$\varphi\equiv\forall v_0(v_0\cdot v_1=v_0)$
and c)
$\varphi\equiv ((v_0+(v_1\cdot v_1)=v_2)\wedge (v_1+v_1\neq v_1))$
The second condition $(v_1+v_1\neq v_1)$ excludes the possibilty of $v_1=0$. Since I add $v_1\cdot v_1$, which is positiv, this formula should state, that $v_0<v_2$
Unfortunatly I am not sure, if I solved this task adequate, and would be thankfull for your opinion.
Tanks in advance.