1. Prove that if A is an uncountable set and B is a countable set then A-B is uncountable.
2. Show that [0,1] is equivalent to (0,1).
Thanks!
If $\displaystyle \scriptstyle A\backslash B$ were countable, then $\displaystyle \scriptstyle (A\backslash B)\cup B$, being the union of two countable sets and a superset of A, would also be countable: a contradiction to the assumption that A is uncountable.
[0,1] is a superset of (0,1), and is also equivalent to the subset [0.5,0.75] of (0,1), hence...2. Show that [0,1] is equivalent to (0,1).