What does this double sided arrow $\longleftrightarrow$ mean?

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.

Solution 1:

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.

Solution 2:

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").