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?
Best Answer
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".