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

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?

• It is often the case (I do because it is better English) that authors DO write "For A to be the case, it is necessary and sufficient, that B should hold as well." So, I would never ever write "if and only if" or the uglier version "iff." Nov 29, 2018 at 4:43
• In my experience, authors always keep the ordering of $A$ and $B$. That is, proof of sufficiency means $A \implies B$ while proof of necessity means $A \impliedby B$. The arbitrariness is only there when you switch the placement of $A$ and $B$.
– Ron
Nov 29, 2018 at 4:48
• I am confused. In the highlighted part you say that the (Necessity) proof is the proof of $A \Rightarrow B$, but below that, you say that that is the (Sufficiency) proof? Nov 29, 2018 at 4:49
• I agree, I have encountered this specific one and has left me confused as well. I don't think that there is a consistent pattern and even bad/wrong English, as @WillM suggests, is constantly reproduced by us non-native English speakers just because we see it in some paper and think that therefore it must also be correct. Nov 29, 2018 at 4:50
• The key point is not which one is "necessary" and which is "sufficient" - if the two words are switched it is not likely to bother many trained mathematicians. Instead, the two words are just used as signposts to make it obvious that both directions of the "if and only if" have been proved. Nov 30, 2018 at 14:40

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 $$B$$".

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.