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The set Q^2 of points in R^2

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October 3, 2010
analytic geometrygraphing linesslopecombinatorial geometryalgebra proposedalgebra

Problem Statement

Consider the set Q2\mathbb Q^2 of points in R2\mathbb R^2, both of whose coordinates are rational.
(a) Prove that the union of segments with vertices from Q2\mathbb Q^2 is the entire set R2\mathbb R^2.
(b) Is the convex hull of Q2\mathbb Q^2 (i.e., the smallest convex set in R2\mathbb R^2 that contains Q2\mathbb Q^2) equal to R2\mathbb R^2 ?
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