Prove or disprove the following statement If A is countably

Prove or disprove the following statement:

If A is countably infinite, B is uncountable, and AB, then the set BA is uncountable.

(Recall that BA={bB bA}.)

Solution

A is a subset of B. A is countably infinite ie A has infinite number of elements and each element can have one to one correspondence with N.

B which is a superset of A is uncountable. This implies B contains elements other than A which are uncountable.

B-A = {bB bA}.)

The set B-A is the set which is the difference of an uncountable set and countable set.

When we remove A the countable set from A, definitely B-A should be uncountable.