Find \(\sigma(u)\), i.e. the \(\sigma\)-algebra generated by \(u\), for the following functions:

\(f, g, h: \mathbb{R} ightarrow \mathbb{R}\),

(i) \(f(x)=x\);

(ii) \(g(x)=x^{2} ;\)

(iii) \(h(x)=|x|\);

\(F, G: \mathbb{R}^{2} ightarrow \mathbb{R}\),

(iv) \(F(x, y)=x+y\)

(v) \(G(x, y)=x^{2}+y^{2}\)

[ under \(f, g, h\) the pre-images of intervals are (unions of) intervals, under \(F\) we get strips in the plane, under \(G\) annuli and discs.]