1. ## Real Analysis

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!

2. Originally Posted by mathgirl1188
1. Prove that if A is an uncountable set and B is a countable set then A-B is uncountable.
If $\scriptstyle A\backslash B$ were countable, then $\scriptstyle (A\backslash B)\cup B$, 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.

2. Show that [0,1] is equivalent to (0,1).
[0,1] is a superset of (0,1), and is also equivalent to the subset [0.5,0.75] of (0,1), hence...