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I'm learning to write mathematical proofs. When the statement to be proven is in the form "$p$ if and only if $q$", the proof is often broken into two parts: necessity and sufficiency. I wonder whether I should organize my proof like:

Necessity: $p \Rightarrow q$

Sufficiency: $ q \Rightarrow p$

... or vice versa?

Since $p \Leftrightarrow q$ is is equivalent to $q \Leftrightarrow p$, does it really matter? Is there any accepted practise to put $p \Rightarrow q$ in necessity or sufficiency, depending on the order in which the statements are presented?

Austin Mohr
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Rashi
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    Regarding the question I think you asked, it does matter. $p\to q$ means $p$ is sufficient for $q$. $q\to p$ means $p$ is necessary for $q$. That's just the English meanings of the words. As to the question I think @Rasmus answered, whether it matters which order you prove $p\to q$ and $q\to p$ in, no, that makes no difference. – Kevin Carlson Oct 18 '12 at 08:40
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    As to why it should matter, if you're in a situation where $q\to p$ is true but $p\leftrightarrow q$ isn't, you might-though it's not all that common-see someone writing sentences like "$p$ is necessary for $q$ to hold" instead of "$q$ implies $p$" or "$p$ holds only if $q$ does". – Kevin Carlson Oct 18 '12 at 08:43
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    Yes, I understand that. What I was really asking was when p<=>q, which condition do we call necessary and which sufficient, since each of them is both? – Rashi Oct 18 '12 at 08:53

1 Answers1

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Strictly speaking, there is no difference, but it is common to put the "subject" first. An example will make more sense.

A subset of $\mathbb{R}^n$ (with the usual topology) is compact if and only if it is closed and bounded.

vs.

A subset of $\mathbb{R}^n$ (with the usual topology) is closed and bounded if and only if it is compact.

They are both the same statement, but the purpose of the theorem is to characterize compactness, not to characterize (closed and bounded)-ness. For this reason, it is more pleasing (I think) to mention compactness first and also to use phrases like "necessary for compactness" and "sufficient for compactness".

Austin Mohr
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