The Symmetric Difference Remains Large: 10543

Solution I by Sung Soo Kim, Hanyang University, Ansan, Kyunggi, Korea. Since S is a union of disjoint intervals whose endpoints are endpoints of the given intervals, S is a finite union of disjoint intervals. When the family has one interval, the length of S is 1; we use induction. With more intervals, let II be a leftmost one, and let I2 be leftmost among the remaiinder. Let S' be the set of points in an odd number of the remaining intervals. We have II I2 C S S' and S' S C I2 II. Since II -I2 and I2 II have the same length, the total length of S is at least that of S', which by the induction hypothesis is at least 1.