What does the two vertical bars mean between a set. So I'm just getting the grasp of set theory and I have this question.

Let $|A| = m$ and $|B| = n$. What is the cardinality of the set $A \times B
$?

I put $\{1,1\}$ as the answer however I wasn't totally sure what the two vertical bars between set $A$ and set $B$ mean. If anyone could clear this up that would be great.
 A: The notation $|A|$ is just a shorthand for "the cardinality of $A$".
A: I suspect you are interpreting the question to be saying that set $A$ contains a single element $m$ and set $B$ contains a single element $n$.  That would explain where the $1$s in your answer are coming from, at least -- although I'm not sure why you think $\{1,1\}$ is the answer.  
But in any case, that is not what it means.  $|A|=m$ means that the set $A$ contains $m$ distinct elements.  $|B|=n$ means that the set $B$ contains $n$ distinct elements.  And $A \times B$ means the set of ordered pairs, where the first element of the pair comes from $A$ and the second element of the ordered pair comes from $B$.
The question you need to ask yourself is:  How many different ways can I fill in the blanks in an ordered pair (_, _) where the first blank is filled in with an element of $A$ and the second blank is filled in with an element of $B$?
A: $|A|$ denotes the cardinality of $A$. Then $|A\times B|=|A|\cdot |B|=mn$ (see Wikipedia).
