Intersection of uncountable set and countable set.

My friend told me that say you have an uncountable set $A$ and a countable set $B$, then the intersection of these two sets is the empty set. But wouldn't something like $A = [0,1]$ and $B = \{1, 2\}$ have the intersection of $\{1\}$? Also is there a way to prove this? Thanks!

• Friends do say some peculiar things.... – Lord Shark the Unknown Oct 10 '17 at 6:04
• Your counterexample is correct. – Qudit Oct 10 '17 at 6:05
• With friends like that, you will never need enemies – Mark Fischler Oct 10 '17 at 6:05
• Yeah I figured he was wrong so does this also mean that the intersection of countable and uncountable set is always countable? Seems like it should be. – aspookyghost20 Oct 10 '17 at 6:07
• Yes, you cannot generate an uncountable set from an intersection (subset) of a countable set. It will be countable or empty. Proof by contradiction – Daniel Oct 10 '17 at 6:43

$A \cap B \subseteq B$.
Hence $A \cap B$ is countable but as you have shown, it need not be empty.
Another counter example would be let $A = \mathbb{R}$ and $B = \mathbb{Q}$, then $A \cap B = \mathbb{Q} \neq \emptyset.$