Using terms "necessary" and "sufficient" when writing proofs for logical equivalence?

Question

Often times in mathematics writing, authors prove the two sides of the implication by showing a proof of necessity and a proof of sufficiency, and the proof takes the following template:

Proposition: $A$ if and only if $B$

Proof.
(Necessity) [... proof of $A \implies B$ here ...]

(Sufficiency) [... proof of $B \implies A$ here ...]

My question is: on what basis does the writer use (Sufficiency) used to mean $B \implies A$ and (Necessity) to mean $A \implies B$? The choice seems rather arbitrary, as one can write the implication $A \implies B$ as either "$A$ is sufficient for $B$" or as "$B$ is necessary for $A$".

Is there a consistent pattern where writers typically choose "sufficient" and "necessary" both in reference to $A$ because $A$ appeared first in the statement of the Proposition?

Answer 1

The traditional reading of "if $A$, then $B$", i.e. $A \to B$, is :

"$A$ is a sufficient condition for $B$" and "$B$ is a necessary condition for $A$".

What happens with :

"$A$ if and only if $B$" ?

It is only a "syntactical" issue. The abbreviation is made of : "$A$ if $B$" and "$A$ only if $B$".

In turn, "$A$ if $B$" is $B \to A$, while "$A$ only if $B$" is $A \to B$.

Thus, from the point of view of natural language, "$A$ if and only if $B$" is :

"$B \to A$ and $A \to B$".

But when "if $B$, then $A$", we say that "$A$ is a necessary condition for $B$", and when "if $A$, then $B$" we also say that "$A$ is a sufficient condition for $B$".

Thus, cooking them together, we read :

"$A$ if and only if $B$"

as :

"$A$ is a necessary and sufficient condition for $B$".

Answer 2

Indeed, when you use these qualifiers, you always refer to the antecedent: "$A$ is necessary" and "$A$ is sufficient". $A$ is the condition you want to prove, $B$ is the given hypothesis.

Think that they are also used in one-way situations. $x>1$ is sufficient for $x>0$, but it is not necessary.