91Ó°ÊÓ

Prove that if \(A \subset B\) and \(A\) is dense in \(B\), then \(\bar{A}=\bar{B}\). Is the statement correct without the assumption that \(A \subset B ?\)

If \(E\) has measure zero, show that the reflected set $$ -E=\\{-x: x \in E\\} $$ also has measure zero.

Is \(\mathbb{R} \backslash \mathbb{Q}\) dense in \(\mathbb{Q} ?\)

Prove that the complement of a dense open subset of \(\mathbb{R}\) is nowhere dense in \(\mathbb{R}\).

Find a calculus textbook proof for the statement that a continuous function \(f\) on an interval \([a, b]\) that has a zero derivative on \((a, b)\) must be constant. Improve the proof to allow a finite set of points on which \(f\) is not known to have a zero derivative.

By altering the construction of the Cantor set, construct a nowhere dense closed subset of \([0,1]\) so that the sum of the lengths of the intervals removed is not equal to 1 . Will this set have measure zero?