Let \(V\) be the set of all pairs \((x,y)\) of real numbers together with the following operations:

\[(x_1,y_1)\oplus (x_2,y_2)= \left(x_{1} x_{2},\,y_{1} y_{2}\right) \]

\[c \odot (x,y) = \left(x^{c},\,y^{c}\right) .\]

(a) Show that there exists an additive identity element, that is:

\[\text{There exists }(w,z)\in V\text{ such that }(x,y)\oplus(w,z)=(x,y). \]

(b) Explain why \(V\) nonetheless is not a vector space.