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?
Best Answer
The traditional reading of "if $A$, then $B$", i.e. $A \to B$, is :
What happens with :
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 :
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 :
as :