Quiz 3 of Math 54400

Quiz 3 (Midterm Exam) Prove that $(A\Delta B)\cap (B\Delta C)\cap (C\Delta A)$ is an empty set. The fractal dimension of a set $E$ on the line is defined as$$d=\lim_{k\to\infty}\frac{\ln N(k)}{\ln k},$$where $N(k)$ is the minimal number of intervals of the length $1/k$ covering the set $E$. Calculating the fractal dimension of the Cantor set and justify your calculation. Let $A$ and $B$ be two closed bounded sets on the line. Prove that the se