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What is $\longleftrightarrow$ used for in mathematics? I know about $\iff$ being used for "If and only if".

Are they the same thing?

I was watching a YouTube video that said:

$$\sum^{\infty}_{n=1} {1\over n^x} \longleftrightarrow \int^{\infty}_{1} {1\over t^x} dt$$

The teacher mentions convergence/divergence, but I was confused when the notation came up.

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Like most notation in math, this is at least somewhat context dependent.... – T. Bongers Jan 18 at 1:42
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depends on context where did you see it? It is possible it is an alternative informal form of iff – Nikos M. Jan 18 at 1:43
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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 Jan 18 at 2:06
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Could you link to the YouTube video in question? – mweiss Jan 18 at 2:49
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For those who want to see the context, the symbol in question appears in the linked YouTube video at about timecode 8:30. The context makes it clear that the intended use is equiconvergence. – mweiss Jan 18 at 3:59
up vote 4 down vote accepted

I understand it as a loose equivalence of convergence, under specific conditions that are not fully mentioned. The presenter writes that a sum diverges/converges if the corresponding integral diverges/converges. It is not an "if and only if", indeed this is not true in general.

The $\leftrightarrow$ symbol appears after the Maclaurin–Cauchy integral test for convergence (the so-called Cauchy integral theorem is quite different). The standard test works under the following conditions:

  • $f$ is continuous, defined on $[n_0, +\infty [$ for some integer $n_0$,
  • $f$ is monotone and decreasing.

Then the infinite series $\sum_{n=n_0}^\infty f(n)$ converges to a finite limit if and only if ($\Leftrightarrow$) the improper integral $\int_N^\infty f(x)\,dx$ is finite. And if the integral diverges, then the series diverges as well. Here, the test works for the $p$-series, as $t \to \frac{1}{t^x}$ is continuous decreasing for $x >0$, and the convergence of the series depends on $x> 1$ or not.

As mentioned in comments, many mathematical symbols have several interpretations (e.g. bijection or logical biconditional).

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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 $$ A \overset{f}{\longleftrightarrow} B $$

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 $\sum\limits_{n=1}^\infty \frac{1}{n^x}$ converges if and only if $\int\limits_{1}^\infty \frac{1}{t^x}dt$ converges.

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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 Jan 18 at 1:51
    
Ahhh see that makes much more sense. – Max Echendu Jan 18 at 2:10
    
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 Jan 18 at 2:11
    
The "bijection" interpretation was unknown to me. Bijection between Specific Elements provides an example of this use (for the other readers) – Laurent Duval Jan 18 at 17:33

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).

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On the one hand, $\longleftrightarrow $ is used for connecting propositional formulas (e.g. $p\to q \lor (p\longleftrightarrow q) \land \lnot w$). 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.

$$\begin{array}{|c|c|c|} \hline p&q&p\longleftrightarrow q\\ \hline T&T&T\\ \hline T&F&F\\ \hline F&T&F\\ \hline F&F&T\\ \hline \end{array}$$

On the other hand, $\iff $ is used as a connective of propositional formulas. You can see both uses here: $$p\longleftrightarrow q \iff (p\to q) \land (q \to p)$$

And what does $a \iff b $ means? If you write $a\iff 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").

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Ahh so it CAN be used as If and only if – Max Echendu Jan 18 at 3:12
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You say that one "is used for connecting propositional formulas" and the other "is used as a connective of propositional formulas". Is one of these meant to have a different word in it, or am I missing the subtle distinction between "for connecting" and "as a connective of"? – Tim Pederick Jan 18 at 4:09
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@TimPederick One is part of the logical language the other is part of the meta-language. – Taemyr Jan 18 at 10:17

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