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?
Best Answer
$A$ 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
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 $B$.