What is the correct notation for equations and sets If i want to continue this equation, which logic connective is the most correct or most people use in tests:
$$ 2x = 4   \rightarrow x= 2       $$ or $$2x =4\Rightarrow x=2    $$ or $$2x=4    \leftrightarrow x=2$$ or $$2x=4\Leftrightarrow x=2$$
or $$2x=4 \equiv x=2  $$
In elementary set theory, i saw many books with this notation:
$$A\subset B\leftrightarrow \forall x\in A\rightarrow x\in B $$ but in this case, if the right-hand side of iff is wrong, all the sentence will be, because we consider the left-hand side as a correct statement. But, with this notation, it will never ocur:
$$A\subset B \equiv \forall x\in A \Rightarrow x\in B$$
Because the right-hand side of iff will always be correct.
 A: The first four are all correct; I definitely would not use the fifth, as $\equiv$ has too many other uses. However, I don’t really recommend that you use any of them: plain English (or whatever language) is preferable. If what’s important to the argument is that you can derive $x=2$ from $2x=4$, then just say something like this:

$2x=4$, so $x=4$.

If what’s important is that the two statements are equivalent, and you’re using that fact to justify some assertion $P$, then say something like this:

$2x=4$ if and only if $x=2$, so $P$.

A: The convention is that the single arrow "$ \rightarrow $" is used as a logical operator in the formal language: $P \rightarrow Q$ is a formula (pronounced "if P then Q") which is defined to be true when $Q$ is true or $P$ is false.
The double arrow "$\Rightarrow$" is conventionally used outside the formal language, as a shorthand for "implies" (or "this means" or "and hence") in the (English) conversation that the writer is having with the reader. So if you're doing a sequence of calculations as part of a proof or exam answer, $\Rightarrow$ is the one you want between each equation.
The two-headed arrows observe the same convention. So "$\leftrightarrow$" is a logical connective, and the formula $P \leftrightarrow Q$ is true if $P$ and $Q$ are both true or both false. You can also write this "$\equiv$".
Thus your set-theory text is using the correct arrows: 
$$A\subset B\leftrightarrow \forall x (x\in A\rightarrow x\in B) $$
is a formula in the language of set theory which defines the symbol $\subset$: since that formula is true, wherever you see $A \subset B$ you can substitute $\forall x (x\in A\rightarrow x\in B)$. (Your alternative notation breaks convention, but if we changed the conventions so that it was O.K, your problems with interpretation would not go away. The new symbols would have the same meaning as the old ones did.)
The dangerous symbol is "$\Leftrightarrow$", which means "implies, and is implied by" or "or equivalently" (or even "if and only if") -- if you use it, it's a claim that you can read your argument up the page as well as down it. There are very few contexts where this is helpful, because most arguments need extra conditions or assumptions somewhere along the way, and using $\Leftrightarrow$ is a standing temptation not to check every stage of the reverse argument. It's much better to write it out twice.
In short, if you need to prove that $P \leftrightarrow Q$, then assume $P$, work until you have proved $Q$ writing $\Rightarrow$ between the lines where appropriate; then assume $Q$ and work (again with $\Rightarrow$s) to prove $P$. And give yourself a big Q.E.D.
A: My intuition as to some symbols is as follows:
$\rightarrow$ is a directional with the connotation "goes to."  So I might write $f(x)\rightarrow 0$ as $x\rightarrow\infty$.
$\Rightarrow$ is an inference which I would say out loud as "implies."  An example would be $ABC$ is a right triangle with right angle at $B\Rightarrow |AB|^2+|BC|^2=|AC|^2$.
$↔$ is not a symbol I've seen used very often, but it suggests a bidirectional "goes to" meaning, which doesn't make sense to me.
$\iff$ is clearly "if and only if" and is the same to me as $⇔$, although the longer arrow is sometimes preferred (by me anyways) to help add readability spacing to a statement.
$\equiv$ is an equivalence relation, like congruence under a modulus.  I have seen it used to mean "is defined as," but that actually added a bit of confusion to what was being said.
Just a note on the right hand side of the example you use at the end:
$$\forall x\in A\Rightarrow x\in B$$
In English, I read this as "for all $x$ in $A$ implies $x$ in $B$."  This doesn't sound right to me, so I would prefer to write the statement as
$$\forall x\in A, x\in B$$
or more directly
$$x\in A\implies x\in B$$
A: There is a solution set to any statement form. We'll denote by “$A(.)\iff B(.)$” that $A(.)$ and $B(.)$ have the same solution set.  In your case: “The double of a number equals $4$” $\iff$ “A number equals $2$.”
