What does $|A|$ denote in set notation? What does $|A|$ of a set $A$ denote?
Also, what does $A\leftrightarrow B$ of sets $A, B$ mean?
I encountered this in one of my textbooks which said:

Of two sets $A, B$ we know $|B|$ but $|A|$ is unknown. If we succeed
  in constructing a bijection $A\leftrightarrow B$, then$|A|=|B|$. A proof which shows
  $|A|=|B|$ by such an explicit construction is called a bijective proof
  or combinatorial proof.

I'm afraid I'm new to set notation. I would be grateful if someone could point me to a good resource on it. 
 A: The notation $|A|$ often means the number of members of the set $A$, the "cardinality".
$A\leftrightarrow B$ might mean that there is a bijection --- a one-to-one correspondence --- between the two sets $A$ and $B$.
A: The notation $|A|$ denotes the cardinal of $A$, this is an abstract (and sometimes more concrete) object which represents the size of $A$. We can think about $|A|$ as an equivalence class in the equivalence relation: $$A\approx B\iff\exists f\colon A\to B\text{ a bijection}$$
Or sometimes we think about $|A|$ as a specific ordinal, which has certain properties and is now called "a cardinal". For example we have finite cardinals (which are the finite ordinals) which represent the natural numbers. So when we say $|\{0,1\}|=2$ we mean that $\{0,1\}$ has two elements. This can be generalized to infinite sets, whose cardinals are called Aleph numbers (denoted by $\aleph$).
A: A bijection means that there is one function that matches every element in A to a different element in B and another function mapping every element in B to a different element in A.  
In small sets say $\{1, 2\}$ and $\{a, b\}$ it is quite simple.
But if you look at "integers" and "even integers" there is a bijection as well, despite the fact that it seems that there are more integers than even integers.
$f_{a->b}(a) = 2a$
$f_{b->a}(b) = b$
In other words, the cardinality of integers is the same as the cardinality of whole even integers.
