Question

If A is an uncountable set andBis a countable set, must A -B be uncountable?.

Short answer

If A is an uncountable set and B is a countable set, A - B needs to be uncountable.

Step-by-step solution

Definition

Definition of Finite, Countable Infinite and Uncountable Infinite.

• A set is finite if it contains a limited number of elements (thus it is possible to list every single element in the set).

• A set is countably infinite if the set contains an unlimited number of elements and if there is a one-to-one correspondence with the positive integers.

• A set is uncountable if the set is not finite or countably infinite.

Proof

Find the A - B is uncountable

Let A be an uncountable set and B be a countable set.

Since $A\cap B\subseteq B,A\cap B$needs to be a countable set as $\left(|\mathrm{A}\cap \mathrm{B}|\le \left|\mathrm{B}\right|\right)$.

When $A\cap B$and A - B would both be countable, then their union $A=(A\cap B)\cup (A-B)$would also be countable (as we could alternately list the elements of and ).

Since we know that A is uncountable while $A\cap B$is countable, we could then require that A - B is uncountable as well.