What is $longleftrightarrow$ used for in mathematics? I know about $iff$ being used for “If and only if“.
Are they the same thing?
UPDATE: Apologies for not giving an example of where I saw it.
I was watching a YouTube video that said:
$$sum^infty_n=1 1over n^x longleftrightarrow int^infty_1 1over t^x dt$$
UPDATE 2: Here is the link to the video.
Note that he’s an excellent teacher, I just didn’t understand it at first but I do now 
http://www.youtube.com/watch?v=ZlYfEqdlhk0
He does mention convergence/divergence, but I was just confused when the notation came up.
Thank you for the informative answers.
On the one hand, $longleftrightarrow $ is used for connecting propositional formulas (e.g. $pto q lor (plongleftrightarrow q) land lnot w$). You can understand it as a binary operator like AND or OR, which are represented by $land $ and $lor $ symbols, as you would know.
Here you can see its truth table.
$$beginarrayc hline p&q&plongleftrightarrow q\ hline T&T&T\ hline T&F&F\ hline F&T&F\ hline F&F&T\ hline endarray$$
On the other hand, $iff $ is used as a connective of propositional formulas. You can see both uses here: $$plongleftrightarrow q iff (pto q) land (q to p)$$
And what does $a iff b $ means? If you write $aiff b $, then you could actually say the same by writing down that the bicondition $a text is true longleftrightarrow b text is true $ is always true. Note that this works whatever the truth values of $a text is true $ or $b text is true$ are.
Edit: in another fields a part of logic, (at least in basic degrees), choosing one or the other does not matter too much ($longleftrightarrow $ or $iff $ are just “lazy” math translations of simple English connector “if and only if”).
In the area of logic, $longleftrightarrow$ is usually used for “if and only if” instead of $iff$ (because who wants to bother drawing that second line all the time).
Otherwise when dealing with functions, $longleftrightarrow$ might also be used to denote a bijective function. So $f colon A leftrightarrow B$ is a bijection between $A$ and $B$. Or you could similarly write $$ A oversetflongleftrightarrow B $$
In regards to what was likely meant in the video that you saw, the following is true:
For a given value of $x$, one has $sumlimits_n=1^infty frac1n^x$ converges if and only if $intlimits_1^infty frac1t^xdt$ converges.
As has been mentioned in the comments, this is almost certainly an idiosyncratic use, and the author (is that the right word for somebody who makes a YouTube video? Probably not) ought to have explained what he or she intended the symbol to mean. Without any additional context, it’s hard to know for sure, but I’m going to hazard a guess that the symbol is intended to denote “are equivalent” in some (perhaps ill-defined) sense. In what sense? Probably in the sense of “equiconvergence” — i.e., their convergence behavior is equivalent (one of them converges if and only if the other one does).
| Like most notation in math, this is at least somewhat context dependent…. – T. Bongers 2 hours ago | ||
Probably. But many symbols do multiple duty. – André Nicolas 2 hours ago | |||
| depends on context where did you see it? It is possible it is an alternative informal form of iff – Nikos M. 2 hours ago | ||
| The use you quote, if it is indeed a full quotation, is a misuse of the symbol that I would be surprised to see used by a professional. It would be OK to say the first converges $longleftrightarrow$ the second converges. – André Nicolas 1 hour ago | ||
| Could you link to the YouTube video in question? – mweiss 56 mins ago |
Ooooohhh it can be used to show a bijection?…Not sure if it's the same case for mine, but love the shared knowledge :)…Thank You. – Max Echendu 1 hour ago | |||
Plus, we were trying to prove divergence of the sum at $x=1$ so this property makes sense. Do you know the name of this property so I can further study it? – Max Echendu 1 hour ago |
