Let $A$ be a finite set and let $B$ be a countably infinite set. What is the cardinality of, (with explanation):
$A - B$ and $B - A$
I couldn't find any similar examples, neither in my professor's book nor on the internet. All I know is that $|B|=|N|$ and $|A| = n$, where $n \in N$.
$A-B $ is a subset of $A $ and as such has a finite cardinality, $|A-B| \le |A|$.
$B-A $ is countably infinite. Assume you know that subsets of $B $ are either finite or countably infinite, then $B-A $ cannot be finite, because then $B=(B-A) \cup (B \cap A) $ would be finite. (The latter is a subset of $A $.)
As for the previous claim (subsets of countably infinite set are either finite or countably infinite), it can be proven by proving on $\mathbb N $: if $S \subset \mathbb N $, choose $s_1=\min S $, $s_2=\min (S-\{s_1\}) $ etc. and this process either exhausts the whole $S $ (in which case $S $ is finite) or provides a map $s:\mathbb N \to S $ which is easily shown to be a bijection, thus $S $ is countably infinite.