In Basic Algebraic Geometry I, exercise 1.6.3 states
Let $X\subset \mathbb{P}^2$ be the reducible $0$-dimensional variety consisting of $3$ points not lying on a line. Prove that the ideal $\mathfrak{U}_X$ cannot be generated by $2$ elements.
Exercise 1.6.5 states
Prove that any finite set of points $S\subset \mathbb{P}^2$ can be defined by two equations.
I am confused by these problems because I thought that for $S$ to be defined by two equations would mean that $\mathfrak{U}_S$ can be generated by 2 elements. However, $X$ in exercise 1.6.3 gives a finite set of points and the problem claims that $\mathfrak{U}_X$ cannot be generated by 2 elements.
Am I misunderstanding something or is one of the problems misstated?
The first exercise is asking you to show that there do not exist $f$ and $g$ so that $(f,g)=\mathfrak{U}_X$.
The second exercise is asking you to find $f,g$ so that $V(f,g)$ and $S$ are equal as sets. This means that you're looking for $f$ and $g$ so that $\sqrt{(f,g)}=\mathfrak{U}_X$. This is different from the first result - the exercises are not in conflict.