Proving algebraic sets i) Let $Z$ be an algebraic set in $\mathbb{A}^n$. Fix $c\in \mathbb{C}$. Show that $$Y=\{b=(b_1,\dots,b_{n-1})\in \mathbb{A}^{n-1}|(b_1,\dots,b_{n-1},c)\in Z\}$$ is an algebraic set in $\mathbb{A}^{n-1}$.
ii) Deduce that if $Z$ is an algebraic set in $\mathbb{A}^2$ and $c\in \mathbb{C}$ then $Y=\{a\in \mathbb{C}|(a,c)\in Z\}$ is either finite or all of $\mathbb{A}^1$. Deduce that $\{(z,w)\in \mathbb{A}^2 :|z|^2 +|w|^2 =1\}$ is not an algebraic set in $\mathbb{A}^2$. 

An idea: for $\,f\in I(Z)\,$ , look at 
$$g(x_1,...,x_{n-1}):=f(x_1,...,x_{n-1},c)\in\Bbb C[x_1,...,x_{n-1}]$$
with $\,c\in\Bbb C\,$ s.t. for some $\,a_1,...,a_{n-1}\,\,,\,(a_1,...,a_{n-1},c)\in Z\,$
