Uncountable Subset A of [0,1] with A – A empty.

December 20, 2011

I’m going through a few books so that I can start doing lots and lots of problems to prepare for my quals.  I’ll be posting some of the “cuter” problems.

Here’s one that, on the surface, looks strange.  But ultimately, the solution is straightforward.

Problem.  Find an uncountable subset $A\subseteq [0,1]$ such that $A - A = \{a - b\,\mid\,\,a,b\in A\}$ has empty interior.

11, 111, 1111, … Not a Square.

December 18, 2011

I just saw this problem in a book of algebra problems, and I thought it was nice given one of the previous posts I had up here.

Problem.  Show that 11, 111, 1111, 11111, … are not squares.

You ought to think about this for a bit.  I started by supposing they were squares and attempting to work it out like that; unfortunately, there are some strange things that happen when we get bigger numbers.  But.  You should see a nice pattern with the last two digits.  Click below for the solution, but only after you’ve tried it!