What does $A\iff B$ mean with respect to "sufficiency" and "necessity"? If I have $A \iff B,$ does that means $A$ is sufficient for $B$ to occurs and $B$ is necessary for $A$ to occurs?
I am confused about the direction of meaning sufficient and meaning necessary, I usually say $\implies$ this direction means the premises is sufficient for the result i.e. $A \implies B$ means $A$ is sufficient for $B$ to occur. And $\Longleftarrow $ means the premises is sufficient for the result i.e. $B \impliedby A$ means $B$ is necessary for $A$ to occur. 
But I have seen some people not sticking to this and taking reverse directions for necessary and sufficient. Could anyone explains this for me please?
 A: If $A \implies B$ then $B$ is necessarily true when $A$ is true. As well, $A$ is a sufficient condition for $B$ to be true.
If $A$ is true then it must be the case that $B$ is true (necessary).
If we want to imply $B$ it is sufficient to determine that $A$ is true.
Imply is a synonym for deduce.
A: $A$ is a sufficient condition for $B$ is symbolised as $$A \to B$$
$A$ is a necessary condition for $B$ is symbolised as $$B \to A$$
Example (taken from the book "forallX" by P.D. Magnus):
Symbolization key:


*

*$P$: Jean is in Paris.

*$F$: Jean is in France.
We can symbolise this sentence

If Jean is in Paris, then Jean is in France.

as $P \to F$.
An intuitive way of understanding the concept of sufficient condition, is: $P$ is a sufficient condition for $F$ means $P$ being true guarantees the truth of $F$. In this example, if Jean is in Paris, I know he is definitely in France.
Along these lines, $F$ is a necessary condition for $P$ means $P$ would not have happened without $F$. In this example, if Jean is not in France, I can be sure he is not in Paris.
$A$ if and only if $B$ is simbolised as $$A \leftrightarrow B$$
It means $A$ is a sufficient and necessary condition for $B$.
A: $A\implies B$ reads that "$B$ is true whenever $A$ is true".
However this does not promise anything about $B$ when $A$ is false.  It is possible that $B$ might be true while $A$ is false, and if so, then $A\implies B$ will still be satisfied.
So $A\implies B$ claims that the truth of $A$ is sufficient but not necessary for the truth of $B$.

Likewise, $A\impliedby B$, also written as $B\implies A$,  claims that the truth of $A$ is necessary but not sufficient for the truth of $B$.

Finally $A\iff B$, being equivalent to $(A\implies B)\wedge(A\impliedby B)$, claims that the truth of each is sufficient and necessary for the truth of the other.
A: 
If I have A⟺B, does that means A is sufficient for B to occurs and B
  is necessary for A to occurs?

Yes it means this, but it means more than this. 
If it only meant this, there would be no difference between bi-implication and simple implication. 
Simple implication ( A $\rightarrow$ B) already means by itself that A is sufficient for B and B is necessary for A. 



*

*Saying that A is sufficient  for B to be true means than $ A \rightarrow B$ is true. ( Which implies, by itself, that B is necessary for A to be true). 

*Saying that A is necessary for B to be true means that if A is not true, B is not true either , that is : $\neg A \rightarrow \neg B$. 

*Now, suppose you want the expression $(A \iff B)$ to mean : " A is both sufficient and necessary for B to be true", you will define this expression as follows : 
$$(A \iff B)\equiv_{Df_1} [(  A \rightarrow B) \land (\neg A \rightarrow \neg B)]$$. 


*

*But, contraposition law ( with double negation) allows you to rephrase the second part of the conjunction as : $(\neg\neg B \rightarrow \neg\neg A) \equiv (B\rightarrow A)$. 


So your first definition is equivalent to : 
$$(A \iff B)\equiv_{Df_{2}} [(  A \rightarrow B) \land (B \rightarrow A)]$$. 
This shows that : saying "A is both sufficient and necessary for B" is equivalent to saying that "(1) A is sufficient for B and (2) B is sufficient for A". 


*

*In fact , with contraposition, one can show that $(A\iff B)$ means equivalently : 


(1) each sentence is sufficient for the other
OR 
(2) each sentence is necessary for the other 
OR 
(3) each is both necessary and sufficient for the other. 
