I need your help to solve this question
A subset $X \subset k^n$ ($k$ a field) is called algebraic if there exist polynomials $f_1, \dots, f_m \in k[t_1,\dots,t_n]$ such that
$X = \{x \in K^n | f_1(x) = ...=f_m(x)=0\}.$
The coordinate ring $k[X]$ of $X$ is the ring of all functions $f: X\to k$ that can be represented by some polynomial. That is, there exists a polynomial $g \in k[t_1,\dots,t_n]$ such that $f(x) = g(x)$ for all $x \in X$.
1- Show that for two different points $x,y \in X$ there exists $f \in k[X]$ with $f(x) \neq f(y)$.
2- Let $k=\mathbb{Z}_2 = \mathbb{Z}/2\mathbb{Z}$ and $X \subset k^2$ be the set $\{(0,1),(1,0)\}$. Is $X$ algebraic? Determine its coordinate ring.
3- Let $g_i$ in $N$ be a sequence of polynomials and define $Y:= \{ x \in k^n | g_i (x) = 0~\forall~i\}$. Show that $Y$ is algebraic set.
Thanks