Two easy questions about algebraic varieties I'm studying Fraleigh's abstract algebra(7ed), and there are little contents about algebraic geometry, just the definitions of varieties and ideals. Since I have few backgrounds about algebraic geometry, I don't know how to solve the following two exercises. 

Sec28, Ex27, b. Give an example of a subset of $\mathbb{R}^2$ which is  not an algebraic variety.
Ex34. Give an example of a subset $S$ of $\mathbb{R}^2$ such that $V(I(S))\neq S$.
  (Here, the algebraic variety $V(S)$ in $F^n$ is the set of all common zeros in $F^n$ of the polynomial in $S$, where $S$ is a finite subset of $F[\mathbf{x}]$.)

I think that the answer of the two exercises can be same. But I don't know how to show some subset is not an algebraic variety. How can I solve it?

Hint1: A univariate polynomial can have infinitely many zeros only if it is the zero polynomial.
Hint2: Show that if the polynomial $p(x,y)$ has infinitely many zeros on the line $y=y_0$, it must be divisible by $y-y_0$.
Hint3: Show that if the conclusion of the previous hint holds for infinitely many choices of $y_0$, then $p(x,y)$ must be the zero polynomial.
