ILL 1983 Problem 5

Consider the set $\mathbb Q^2$ of points in $\mathbb R^2$ , both of whose coordinates are rational.(a) Prove that the union of segments with vertices from $\mathbb Q^2$ is the entire set $\mathbb R^2$ .(b) Is the convex hull of $\mathbb Q^2$ (i.e., the smallest convex set in $\mathbb R^2$ that contains $\mathbb Q^2$ ) equal to $\mathbb R^2$ ?