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When writing papers on maths, when is it appropriate to use \rightarrow or \leftrightarrow, and when should one refrain from using them. For example, would this enter image description here

be appropriate use of \leftrightarrow.

EDIT: What about this enter image description here Credit to Servaes, Peter Woolfitt and Michael, not trying to steal or anything, just using as an example. Should I make some arrows, or should I just line them up like in the picture.

redelectrons
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Arrows such as $\iff$ and $\implies$ mean "if and only if" and "implies" and are used to show connection between statements. In the case of what you wrote you simply need equalities "$=$".

And arrows such as $\rightarrow$ have even different meanings. This one I see most often used to specify a domain and codomain of a function, like this: $f: \mathbb{R} \rightarrow \mathbb{Z}$. But it probably has other uses too.

Kuba
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    $\to$ also means "syntactically implies" in logic. For example, $\vdash p \to (q \to p)$. – Patrick Stevens Nov 28 '15 at 22:52
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    Yes, in logic, some authors distinguish between single arrows and double arrows. e.g. one might refer to the material conditional--which is what the example you gave, @PatrickStevens, probably means--and the other might mean something stronger, such as logical implication. In general, I think you have to know the context. Often, a careful author will define his/her arrows. In other cases, you can just figure it out if you understand the material well enough. – Mars Nov 28 '15 at 22:59