What is the meaning of the notation $A :\Leftrightarrow B$? This is the text from my book:


  To define a statement $A$ so that it is true whenever the statement
  $B$ is true, we write $$A :\Leftrightarrow B$$ and say '$A$ is true,
  by definition, if $B$ is true'.


I don't quite understand it. How does it differ from $A := B$ or $A \Leftrightarrow B$? Please provide some examples if you can? Thanks in advance.

Edit:

Just to check whether I have really understood this notation, is it correct to write $(X \subseteq Y) :\Leftrightarrow (\forall x \in X (x \in Y))$?
$A:=B$ means that $A$ is defined as $B$. 

$A \Leftrightarrow B$ is a theorem: it means that $A$ and $B$ are defined independently and now you are saying that indeed $A$ holds if and only if $B$ holds. This should be followed by some proof.

$A :\Leftrightarrow B$ is again a definition. It means that this is not something you have to prove (you may need to prove that this is a good definition, but that's another problem).

Examples:

$f(x):= x^3$ : definition. Nothing to prove

$f(x)&gt;0 \Leftrightarrow x&gt;0$ : theorem, you should provide a proof for it (in this case it's trivial)

$n$ is even $:\Leftrightarrow n = 2k$ for some $k \in \mathbb{N}$: definition of being even. Nothing to prove

Edit: yes, your example is correct as that's actually the definition of being a subset
One possible reading is that A may be true when B is false, in contrast to "A if and only if B" meaning not only that A is true whenever B is, but that A is false whenever B is.