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construct A dense in [0,1]^2 and line intersections

Source: VJIMC 2007 2.1

June 24, 2021
set theorydense

Problem Statement

Construct a set A⊂[0,1]×[0,1]A\subset[0,1]\times[0,1] such that AA is dense in [0,1]×[0,1][0,1]\times[0,1] and every vertical and every horizontal line intersects AA in at most one point.
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